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evala/Minpoly

minimal polynomial of an algebraic number (or function)

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

evala(Minpoly(a, x, K))

Parameters

a

-

algebraic number or function

x

-

name

K

-

(optional) set of algebraic numbers or functions defining an extension field

Description

• 

The Minpoly function is a placeholder for representing the minimal polynomial of an algebraic number (or function) a. It is used in conjunction with evala.

• 

The call evala(Minpoly(a, x)) computes the monic minimal polynomial of a in the variable x over the field of rational numbers (or multivariate rational functions). The resulting polynomial will not contain any algebraic numbers or functions.

• 

The call evala(Minpoly(a, x, K)) computes the monic minimal polynomial of a in the variable x over the field F resulting from an extension of the rational numbers (or multivariate rational functions) by the algebraic numbers (or functions) in K. The resulting polynomial will only contain algebraic numbers or functions from F in its coefficients.

• 

The variable x cannot occur in either a or K; otherwise, an error will be raised.

• 

The algebraic numbers and functions in both a and K can be given either in radical or RootOf notation. A mixture or radicals and RootOfs is not supported. The coefficients of the resulting polynomial will be returned in the same form as K (if specified).

• 

If the algebraic numbers and functions in K do not form a syntactical subset of the algebraic numbers and functions occurring in a, evala/Algfield will be used to rewrite a as an element of an appropriate extension field of F. This may not always succeed, and as a result, the polynomial returned may not be of minimal degree in that case.

Examples

aliassqrt2=RootOfx22,index=1:

aliasi=RootOfx2+1,index=1:

aliassqrty=RootOfx2y,x:

evalaMinpoly1+sqrt2,x

x22x1

(1)

evalaMinpolysqrt2+sqrty,x

x42y+2x2+y24y+4

(2)

evalaMinpolysqrt2sqrty,x

x22y

(3)

Specifying an extension field.

aliasα=RootOfx4+1,index=1:

evalaMinpolyα,x,i

x2i

(4)

evalaMinpolyα,x,sqrt2

sqrt2x+x2+1

(5)

Using radical instead of RootOf notation:

evalaMinpoly114,x,I

x2I

(6)

fevalaMinpoly114,x,sqrt2

f2x+x2+1

(7)

evalf,x=114

2−114+1+I

(8)

radnormal

0

(9)

If the algebraic numbers and functions are not independent, i.e., they do not form a field, or if the algebraic numbers and functions in K do not occur in a, the resulting polynomial may not be of minimal degree. In the following example, Maple is unable to tell whether β2 equals sqrt2 or -sqrt2.

aliasβ=RootOfx42:

evalaMinpolyβ,x,sqrt2

x42

(10)

In this example, using an indexed RootOf will help.

evalaMinpolyRootOfx42,index=1,x,sqrt2

x2sqrt2

(11)

evalaMinpolyRootOfx42,index=2,x,sqrt2

x2+sqrt2

(12)

Compatibility

• 

The evala/Minpoly command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

Algfield

evala

Norm

PolynomialTools[MinimalPolynomial]

RootOf