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.

$\mathrm{with}\left(\mathrm{diffalg}\right)\:$

$\mathrm{K0}≔\mathrm{field\_extension}\left(\mathrm{transcendental\_elements}=\left[a\right]\right)$

$\mathrm{K0}≔\mathrm{ground\_field}$

$\mathrm{K1}≔\mathrm{field\_extension}\left(\mathrm{relations}=\left[ab-1\,c-d\right]\,\mathrm{base\_field}=\mathrm{K0}\right)$

$\mathrm{K1}≔\mathrm{ground\_field}$

$\mathrm{R0}≔\mathrm{differential\_ring}\left(\mathrm{field\_of\_constants}=\mathrm{K1}\,\mathrm{derivations}=\left[x\right]\,\mathrm{ranking}=\left[u\right]\right)$

$\mathrm{R0}≔\mathrm{ODE\_ring}$

$p≔a{b}^{2}u\left[x,x\right]+c{u\left[\right]}^2+d^3{u\left[x\right]}^3+1$

$p≔d^3{u}_x^3+a{b}^2{u}_{x,x}+c{u\left[\right]}^2+1$

$\mathrm{reduced\_form}\left(p\,\mathrm{R0}\right)$

$d^3{u}_x^3+a{b}^2{u}_{x,x}+d{u\left[\right]}^2+1$

$P≔\mathrm{Rosenfeld\_Groebner}\left(\left[acu\left[x\right]-4d{u\left[\right]}^2\right]\,\mathrm{R0}\right)$

$P≔\left[\mathrm{characterizable}\right]$

$\mathrm{equations}\left(P\right)$

$\left[\left[a{u}_x-4{u\left[\right]}^2\right]\right]$

$\mathrm{K2}≔\mathrm{field\_extension}\left(\mathrm{prime\_ideal}=P\right)$

$\mathrm{K2}≔\mathrm{ground\_field}$

$\mathrm{K3}≔\mathrm{field\_extension}\left(\mathrm{transcendental\_elements}=\left[e\right]\,\mathrm{base\_field}=\mathrm{K2}\right)$

$\mathrm{K3}≔\mathrm{ground\_field}$

$\mathrm{R1}≔\mathrm{differential\_ring}\left(\mathrm{field\_of\_constants}=\mathrm{K3}\,\mathrm{derivations}=\left[y\right]\,\mathrm{ranking}=\left[v\right]\right)$

$\mathrm{R1}≔\mathrm{ODE\_ring}$

$q≔\frac{\left(au\left[x,x\right]-8u\left[\right]u\left[x\right]\right)v\left[y\right]+\left(b+e\right)v\left[y,y\right]}{u\left[x\right]+x}$

$q≔\frac{\left(a{u}_{x,x}-8u\left[\right]u_x\right)v_y+\left(b+e\right)v_{y,y}}{u_x+x}$

$\mathrm{reduced\_form}\left(q\,\mathrm{R1}\right)$

$\frac{v_{y,y}b+v_{y,y}e}{u_x+x}$