LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QUADRATIC FIELDS and CLASS NUMBER FORMULA Miriam Ciavarella Universit\303\240 degli Studi di Torino Italy miriam.ciavarella@unito.it Marina Marchisio Universit\303\240 degli Studi di Torino Italy marina.marchisio@unito.it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The aim of this document is to give same procedures in order to work explicitely with quadratic fields; in particular the idea of work was born in order to find a useful procedure to compute the class number of a quadratic filed. Many problems of number theory lead to another important question in the arithmetic of algebraic number fields, the question of decomposition of algebraic numbers into prime factors. We shall define a procedure Dec which returns the decomposition of algebraic numbers into prime factors in a quadratic filed. The problems of factorization are very closely connected with Fermat's (last) theorem. Historically, it was precisely the problem of Fermat's theorem which led Kummer to his fundamental work on the arithmetic of algebraic numbers. The number of divisor classes is a characteristic of the set of all divisors of the field. It is well known the important role of the number h of divisor classes of algebraic number filed play in the arithmetic of the field. Thus one would like to have an explicit formula for the number h in terms of simpler values which depend on the filed. Although this has not been accomplished for arbitrary algebraic number fields, for certain fields of great interest, such as quadratic fields, such formulas as been found. Since all divisors are products of prime divisors and the number of prime divisors is infinite, then to compute the number h in a finite number of steps we must use some infinite processes. This is why, in the determination of h, we shall have to consider infinite products, series and other analytic concepts. Notation: In the sequel of this work we shall use the following notation: Q the field of rational numbers; Z the ring of integer numbers. JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= A quadratic field is an algebraic number field K of degree two over Q. Any quadratic field has the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZCLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRidGK0YrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZFbGVtZW50O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRIlFGJ0YvRjJGOQ== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbXNxcnRHRiQ2Iy1GIzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSgmbm90aW47RidGOi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYxNiVRIlFGJ0Y0RjdGOg==. 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 if and only if 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZFbGVtZW50O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRIlFGJ0YvRjJGOQ==. Therefore any quadratic field has the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a square-free integer that is uniquely determined by the field. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC is called a real and when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= an imaginary, quadratic field.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We give a procedure QuadraticField for writing the quadratic field for any integer (not necessarily square-free). Input datum is d (not necessarily square-free) an integer numberLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRidGK0YrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==. Output datum is the quadratic field determined by d. JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUS9RdWFkcmF0aWNGaWVsZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEjMTJGJy9GOFEnbm9ybWFsRidGQ0ZDRkNGKy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRkM=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEvUXVhZHJhdGljRmllbGRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEjMTJGJ0Y+LyUrZXhlY3V0YWJsZUdGQkY+Rj5GWEY+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYuKipJJFRoZUc2IiIiIkkqcXVhZHJhdGljRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmRidJJmZpbGVkR0YmRidJI2lzR0YpRidGJy1JIlFHRiY2IyomXiNGJ0YnKSIiJCNGJyIiI0YnRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is related to the volume of the fundamental domain of the ring of integers, and it regulates which primes are ramified. The discriminant is one of the most basic invariants of a number field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, and the analytic class number formula for K. An old theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, and the subject of current research. Let K be an algebraic number field, and let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU= be its ring of integers. Let 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 be an integral basis of LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtRiw2JVEiS0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== (i.e. a basis as a Z-module), and let 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 be the set of embeddings of K into the complex numbers (i.e. ring homomorphisms K\342\206\222C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYvNiVRImlGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlLUYvNiVRImJGJ0Y9Rj8tRiM2JC1GLzYlUSJqRidGPUY/RjVGQUY1RjVGNQ==.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The procedure DiscriminantQ computes the discriminant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1GLDYjUSFGJ0ZC. Input datum is d (not necessarily square-free) a integer number which defines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC. Output datum is the discriminant of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1GLDYjUSFGJ0ZC. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEuRGlzY3JpbWluYW50UUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUkjbW5HRiQ2JFEjMTJGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUScmc2RvdDtGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlUtRjs2JFEiN0YnRj4vJStleGVjdXRhYmxlR0ZGRj5GPkZlbkY+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiNA

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

ISIl

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The ring of integers of an algebraic number field K, often denoted by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ==, is the ring of algebraic integers contained in K.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We give a procedure RingInteger which compotes the ring of integers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC. Input datum is d (not necessarily square-free) a integer number which defines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC. Output datum is the ring of integers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1GLDYjUSFGJ0ZC. 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print(); # input placeholder

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqKi4qKkkkVGhlRzYiIiIiSShpbnRlZ2VyRyUqcHJvdGVjdGVkR0YoSSVyaW5nR0YnRihJI29mR0YnRihGKC1JIlFHRic2IyomXiNGKEYoKSIiJiNGKCIiI0YoRiguSSNpc0c2JEYqSShfc3lzbGliR0YnRihJIlpHRidGKEYoKihGMUYoRjJGKEY6RihGKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEsUmluZ0ludGVnZXJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIzIwRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEnJnNkb3Q7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIjNGJ0Y+LyUrZXhlY3V0YWJsZUdGRkY+Rj5GZW5GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYqKi4qKkkkVGhlRzYiIiIiSShpbnRlZ2VyRyUqcHJvdGVjdGVkR0YoSSVyaW5nR0YnRihJI29mR0YnRihGKC1JIlFHRic2IyokKSIjOiNGKCIiI0YoRiguSSNpc0c2JEYqSShfc3lzbGliR0YnRihJIlpHRidGKEYoKiZGMUYoRjlGKEYo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The units of a number filed K are the invertible elements of the ring of integers. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJRRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtc3FydEdGJDYjLUYjNiQtRiw2JVEiZEYnRi9GMkY5RjlGOUY5 is a real quadratic filed then the positive units form a multiplicative group which is isomorphic to Z. This group admits a generator >1; this generator is called fundamental unit of K. Let we recall that in the case of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZlcXVpdjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctRjM2L1EkbW9kRicvJSVib2xkR1EldHJ1ZUYnL0YwUSVib2xkRicvJStmb250d2VpZ2h0R0ZRRjZGOUY7Rj1GP0ZBRkNGRUZIRjItRiw2JFEiNEYnRi9GMkYv the units of K are the natural integer solutions of the Pell-Fermat equation; thus if 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 is the fundamental unit of K and we put 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then the sequence 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 gives all the solutions of the Pell-Fermat equation.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We give a procedure FU which computes the fundamental unit of a real quadratic field. Input datum is d (not necessarily square-free) a positive integer number which defines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC. Output datum is the fundamental unit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1GLDYjUSFGJ0ZC. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjRlVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1JI21uR0YkNiRRIzMzRicvRjNRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPkY+RkBGPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYiI0IiIiIqJiIiJUYkKSIjTCNGJCIiI0YkRiQ=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjRlVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIzMzRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEnJnNkb3Q7RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIjRGJ0Y+LyUrZXhlY3V0YWJsZUdGRkY+Rj5GZW5GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCYiI0IiIiIqJiIiJUYkKSIjTCNGJCIiI0YkRiQ=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Any prime number p gives rise to an ideal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcE9GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GLzYlUSJLRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USdub3JtYWxGJw== in the ring of integers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== of a quadratic field K. In line with general theory of splitting of prime ideals in Galois extensions, this may be: p is inert : (p) is a prime ideal; the quotient ring is the finite field with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= elements: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== /pLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUklbXN1cEdGJDYlLUYvNiVRInBGJ0YyRjUtRiM2JS1JI21uR0YkNiRRIjJGJy9GNlEnbm9ybWFsRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGSkZG. p splits : (p) is a product of two distinct prime ideals of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== ; the quotient ring is the product LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== /pLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRInBGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSgmdGltZXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRldGK0ZE. p is ramified : (p) is the square of a prime ideal of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiT0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJLRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== ; the quotient ring contains non-zero nilpotent elements. The third case happens if and only if p divides the discriminant D. The first and second cases occur when the Kronecker symbol equals -1 and +1, respectively. The first two cases are in a certain sense equally likely to occur as p runs through the primes, see Chebotarev density theorem. The law of quadratic reciprocity implies that the splitting behaviour of a prime p in a quadratic field depends only on p modulo D, where D is the field discriminant.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We give a procedure Dec which returns how each rational prime p factors as a product of prime divisors in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC. Input data are d (not necessarily square-free) a positive integer number which defines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC and p a prime number. Output datum is the decomposition of the prime p in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1GLDYjUSFGJ0ZC. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZmXmwtSSNtaUdGJDYlUSREZWNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYvUSVwcm9jRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZVRjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GWS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRImRGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZYL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInBGJ0YvRjJGOS1GLDYjUSFGJy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0ZecC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYvUSZsb2NhbEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGWEZaLUYsNiVRIm1GJ0YvRjJGXW8tRiw2JVEiaUYnRi9GMkZdby1GLDYlUSJhRidGL0YyLUY2Ni1RIjtGJ0Y5RjtGYG9GQEZCRkRGRkZIRlhGTUZpb0ZqcC1GNjYvUSNpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRmpwRmpuLUY2Ni1RIj5GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkjbW5HRiQ2JFEiMEYnRjlGanAtRjY2L1EldGhlbkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRmpwLUY2Ni9RJGZvckYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRmpwRmBxRmpwLUY2Ni9RJWZyb21GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZqcC1GYHI2JFEiMUYnRjlGanAtRjY2L1EjdG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZqcEZqbkZqcC1GNjYvUSNkb0YnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRmpwRmlvRmpwRmlxRmpwLUYsNiVRJXR5cGVGJ0YvRjItRmZuNiQtRiM2JS1JJm1mcmFjR0YkNihGam4tRiM2Iy1JJW1zdXBHRiQ2JUZgcS1GIzYjLUZgcjYkUSIyRidGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmF1LyUpYmV2ZWxsZWRHRj1GXW8tRiw2JVEoaW50ZWdlckYnRi9GMkY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVGMUYvRjJGanBGY3JGanBGY3FGNUZcdEZqcC1GNjYvUSRlbmRGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZqcEZpcUZmcUZqcEZedkZqcEZic0ZmcUZqcEZedkZqcEZpcUZmcUZpb0ZqcEZpcUZqcEZqbi1GNjYtUSI8RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZfckZqcEZjckZqcEZdcUY1LUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZodkZqbkZmcUZqcEZmckZqcEZgcUZqcEZpckZqcEZmdEZqcEZfc0ZqcEZdcUZqcEZic0ZqcEZpb0ZqcEZpcUZqcEZlcy1GZm42JC1GIzYlLUZddDYoLUYjNiNGXXFGX3RGXHVGX3VGYnVGZHVGXW9GZnVGOUZpdUZcdkZqcEZjckZqcEZdcUY1Rl53Rl52RmpwRmlxRmZxRmpwRmNxRjVGZHZGXXEtRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GXnZGanBGYnNGZnFGanBGXnZGanBGaXFGZnFGanBGanBGanBGaW8tRmpvNiZGXHBGX3BGYnAvRmVwUSVhdXRvRidGaXFGanBGZXMtRmZuNiQtRiM2JUZjb0Zdby1GLDYlUSZwcmltZUYnRi9GMkY5Rml1LUYsNiVGPUYvRjJGanBGY3JGanAtRiw2JVEmcHJpbnRGJ0YvRjItRmZuNiQtRiM2MS1GNjYtUSInRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRlotRiw2JVEkVGhlRicvRjBGPUY5RmpwLUYsNiVRJ251bWJlckYnRmF5RjlGanBGY29GanAtRiw2JVElbXVzdEYnRmF5RjlGanAtRiw2JVEjYmVGJ0ZheUY5RmpwLUYsNiVGZXFGYXlGOUZqcC1GLDYlRl94RmF5RjlGaXhGOUZqcEZedkZqcEZpcUZmcUZqcEZpb0Zldy1GLDYlUSV3aXRoRidGL0YyLUZmbjYkLUYjNiMtRiw2JVEqbnVtdGhlb3J5RidGL0YyRjlGYndGaW9GanBGaXFGanBGY28tRjY2LVErJk5vdEVxdWFsO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GZnRGanAtRjY2L1EkYW5kRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGanBGanAtRiw2JVEpbGVnZW5kcmVGJ0YvRjItRmZuNiQtRiM2JUZjcUZdb0Zjb0Y5Rml1RlxzRmpwRmNyRmpwRmJ4LUZmbjYkLUYjNi1GaXhGXnlGanBGXXpGanBGY29GanAtRiw2JVEnc3BsaXRzRidGYXlGOUZpeEZmb0Zmb0Y5RmpwRl52RmpwRmlxRmZxRmlvRmpwRmlxRmpwRmNvRml6RmZ0RmpwRlxbbEZqcEZqcEZfW2xGYltsRml1RmR2RlxzRmpwRmNyRmpwRmJ4LUZmbjYkLUYjNiZGZm8tSSNtc0dGJDYjUTVUaGV+cHJpbWV+cH5pc35pbmVydEYnRmZvRmZvRjlGanBGXnZGanBGaXFGZnFGaW9GanBGaXFGanBGY29GaXpGZnRGanBGXFtsRmpwRmpwRmVzLUZmbjYkLUYjNiUtRl10NigtRiM2I0ZjcS1GIzYjRmNvRlx1Rl91RmJ1RmR1Rl1vRmZ1RjlGaXVGXHZGanBGY3JGanBGYngtRmZuNiQtRiM2JUZmby1GYlxsNiNROFRoZX5wcmltZX5wfmlzfnJhbWlmaWVkRidGZm9GOUZqcEZedkZqcEZpcUZmcUZpb0ZqcEZpcUZqcEZjb0ZpdUZmdEZqcEZjckZpb0ZqcEZpcUZqcC1GLDYlUSVtb2RwRidGL0YyLUZmbjYkLUYjNiVGY3FGXW8tRmByNiRRIjhGJ0Y5RjlGaXVGXHNGanBGY3JGanBGYngtRmZuNiQtRiM2JUZmby1GYlxsNiNRKTJ+c3BsaXRzRidGZm9GOUZqcEZedkZqcEZpcUZmcUZpb0ZqcEZpcUZqcEZmXWxGaV1sRml1LUZgcjYkUSI1RidGOUZqcEZjckZqcEZieC1GZm42JC1GIzYlRmZvLUZiXGw2I1ErMn5pc35pbmVydEYnRmZvRjlGanBGXnZGanBGaXFGZnFGaW9GZXdGaXFGanBGZl1sLUZmbjYkLUYjNiVGY3FGXW8tRmByNiRRIjRGJ0Y5RjlGaXVGZnRGanAtRjY2L1Ejb3JGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZqcEZmXWxGYV9sRml1LUZgcjYkUSIzRidGOUZqcEZqcEZjckZqcEZieEZgXmxGanBGXnZGanBGaXFGZnFGanBGXnZGanBGaXFGYndGaW9GanBGXnZGanBGTw==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

UTVUaGV+cHJpbWV+cH5pc35pbmVydDYi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkRGVjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictSSNtbkdGJDYkUSMxNUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjNGJ0Y+LyUrZXhlY3V0YWJsZUdGRkY+Rj5GZm5GPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

UThUaGV+cHJpbWV+cH5pc35yYW1pZmllZDYi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The type of decomposition of a prime number p depends only on its residue modulo |D| , so that all prime numbers having the same residue have the same decomposition. The Dirichlet character \317\207 of the filed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC makes this form of the decomposition rule more clear. The character \317\207 is a Dirichlet character modulo |D| where D is the discriminant of the filed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1GLDYjUSFGJ0ZC and the decomposition of a prime p in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC is given the conditions: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5Njc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInBGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjFGMkYyLUkjbW9HRiQ2LVEiPUYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkjbW5HRiQ2JFEiMUYnRjJGQUYy; 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEnJiM5Njc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInBGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjFGMkYyLUkjbW9HRiQ2LVEiPUYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUZENi1RKiZ1bWludXMwO0YnRjJGR0ZJRktGTUZPRlFGUy9GVlEsMC4yMjIyMjIyZW1GJy9GWUZobi1JI21uR0YkNiRRIjFGJ0YyRkFGMg==; 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if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5Njc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInBGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjFGMkYyLUkjbW9HRiQ2LVEiPUYnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkjbW5HRiQ2JFEjMC5GJ0YyRkFGMg== JSFH This character is fundamental for the computation of the class number of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We now give a procedure Car which gives the Dirichlet character \317\207 of the filed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC . LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Note that Car procedure needs DiscriminantQ procedure to be clicked. Input data are d (not necessarily square-free) a positive integer number which defines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC and x an integer number. Output datum is the value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5Njc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== . 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

LUkkQ2FyRzYiNiQiIzoiIiQ=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIh

LUkkQ2FyRzYiNiQiI0AiIiU=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If this group is finite, (as it is in the case of the ring of integers of a number field) then the order of the group is called the class number. The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be the an algebraic number field; let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEoJiM4NDY1O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJPRidGL0YyLUYjNiUtRiw2JVEiS0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRkVGRy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGMkYy be the group of the fractional ideals of K and H be the group of the principal fractional ideals of K. Then 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 and the class number is 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. The computation of the class number is closely related to the Dedekind zeta function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIktGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRInNGJ0Y9Rj9GNUY1RjU= of K, in particular it is related to the residue in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NTA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIktGJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYvNiVRInNGJ0Y9Rj9GNUY1RjU=. If K is a quadratic field, the Dedekind zeta function is the product 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJnpldGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== is the zeta Riemann function and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInNGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJiZjaGk7RicvRjBGRUZBRkFGQUZB is the Dirichlet L function. Thus the number h is related to the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJyYjOTY3O0YnL0YwRkZGPkY+Rj5GPg== which can be explicitely computed.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= We give a procedure ClassNumber which computes the class number of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC. This procedure needs the DiscriminantQ, Car, FU procedure to be clicked. Input datum is d (not necessarily square-free) a positive integer number which defines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQkZC . Output datum is the class number of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkmbXNxcnRHRiQ2Iy1GIzYkLUYsNiVRImRGJ0YvRjIvRjNRJ25vcm1hbEYnRkJGQi1JI21vR0YkNi1RIi5GJ0ZCLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRkovJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRllGQg== JSFH JSFH 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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[1] Z.I.Borevich and I.R.Shafarevich, Number Theory, Academic Press, Inc., 1966 [2] P.Samuel, Th\303\251orie Alg\303\251brique des Nombres, Hermann, Editeurs des Sciences et des Arts, 1971.

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