diffalg(deprecated)/field_extension - Maple Help

diffalg

 field_extension
 define a field extension of the field of the rational numbers

 Calling Sequence field_extension (transcendental_elements = L, base_field = G) field_extension (relations = J, base_field = G) field_extension (prime_ideal = P)

Parameters

 L - list or set of names G - (optional) ground field J - list or set of polynomials P - characterizable differential ideal

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function field_extension returns a table representing a  field extension of the field of the rational numbers. This field can be used as a field of constants for differential polynomial rings.
 • For all the forms of field_extension, the parameter base_field = G can be omitted. In that case, it is taken as the field of the rational numbers.
 • The first form of field_extension returns the purely transcendental field extension $G\left(L\right)$ of G.
 • The second form of field_extension returns the field of the fractions of the quotient ring G [X1 ... Xn] / (J) where the Xi are the names that appear in the polynomials of R and do not belong to G and (J) denotes the ideal generated by J in the polynomial ring G [X1 ... Xn].
 You must ensure that the ideal (J) is prime, field_extension does not check this.
 • The third form of field_extension returns the field of fractions of R / P where P is a characterizable differential ideal in the differential polynomial ring R.
 You must ensure that the characterizable differential ideal P is prime. The function field_extension does not check this.
 The embedding differential polynomial ring of P must be endowed with a jet notation.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $\mathrm{K0}≔\mathrm{field_extension}\left(\mathrm{transcendental_elements}=\left[a\right]\right)$
 ${\mathrm{K0}}{≔}{\mathrm{ground_field}}$ (1)
 > $\mathrm{K1}≔\mathrm{field_extension}\left(\mathrm{relations}=\left[ab-1,c-d\right],\mathrm{base_field}=\mathrm{K0}\right)$
 ${\mathrm{K1}}{≔}{\mathrm{ground_field}}$ (2)
 > $\mathrm{R0}≔\mathrm{differential_ring}\left(\mathrm{field_of_constants}=\mathrm{K1},\mathrm{derivations}=\left[x\right],\mathrm{ranking}=\left[u\right]\right)$
 ${\mathrm{R0}}{≔}{\mathrm{ODE_ring}}$ (3)
 > $p≔a{b}^{2}u\left[x,x\right]+c{u\left[\right]}^{2}+{d}^{3}{u\left[x\right]}^{3}+1$
 ${p}{≔}{{d}}^{{3}}{}{{u}}_{{x}}^{{3}}{+}{a}{}{{b}}^{{2}}{}{{u}}_{{x}{,}{x}}{+}{c}{}{{u}\left[\right]}^{{2}}{+}{1}$ (4)
 > $\mathrm{reduced_form}\left(p,\mathrm{R0}\right)$
 ${{d}}^{{3}}{}{{u}}_{{x}}^{{3}}{+}{a}{}{{b}}^{{2}}{}{{u}}_{{x}{,}{x}}{+}{d}{}{{u}\left[\right]}^{{2}}{+}{1}$ (5)
 > $P≔\mathrm{Rosenfeld_Groebner}\left(\left[acu\left[x\right]-4d{u\left[\right]}^{2}\right],\mathrm{R0}\right)$
 ${P}{≔}\left[{\mathrm{characterizable}}\right]$ (6)
 > $\mathrm{equations}\left(P\right)$
 $\left[\left[{a}{}{{u}}_{{x}}{-}{4}{}{{u}\left[\right]}^{{2}}\right]\right]$ (7)
 > $\mathrm{K2}≔\mathrm{field_extension}\left(\mathrm{prime_ideal}=P\right)$
 ${\mathrm{K2}}{≔}{\mathrm{ground_field}}$ (8)
 > $\mathrm{K3}≔\mathrm{field_extension}\left(\mathrm{transcendental_elements}=\left[e\right],\mathrm{base_field}=\mathrm{K2}\right)$
 ${\mathrm{K3}}{≔}{\mathrm{ground_field}}$ (9)
 > $\mathrm{R1}≔\mathrm{differential_ring}\left(\mathrm{field_of_constants}=\mathrm{K3},\mathrm{derivations}=\left[y\right],\mathrm{ranking}=\left[v\right]\right)$
 ${\mathrm{R1}}{≔}{\mathrm{ODE_ring}}$ (10)
 > $q≔\frac{\left(au\left[x,x\right]-8u\left[\right]u\left[x\right]\right)v\left[y\right]+\left(b+e\right)v\left[y,y\right]}{u\left[x\right]+x}$
 ${q}{≔}\frac{\left({a}{}{{u}}_{{x}{,}{x}}{-}{8}{}{u}\left[\right]{}{{u}}_{{x}}\right){}{{v}}_{{y}}{+}\left({b}{+}{e}\right){}{{v}}_{{y}{,}{y}}}{{{u}}_{{x}}{+}{x}}$ (11)
 > $\mathrm{reduced_form}\left(q,\mathrm{R1}\right)$
 $\frac{{{v}}_{{y}{,}{y}}{}{b}{+}{{v}}_{{y}{,}{y}}{}{e}}{{{u}}_{{x}}{+}{x}}$ (12)