The Subfields function is a placeholder for representing a primitive description of an algebraic extension. It is used in conjunction with evala.

Let f be an irreducible polynomial in K[x]. If f contains only one variable then x need not be specified, otherwise both K and x must be specified. If the argument K is not specified then K is the smallest extension of the rationals such that the coefficients of f are in K. If K is specified then the field K contains the RootOfs in this set as well. Let L be the field extension of K given by one single root of f. So L is not the splitting field; L = K[x]/(f) = K(RootOf(f,x). The call evala(Subfields(f, deg, K, x)) computes the set of all subfields of L over K of degree deg. Each subfield is given by a single RootOf of degree deg.

A field K(R) where R is a RootOf is a subfield of L if and only if f has an irreducible factor g over K(R) such the degree of f equals the product of the degree of g and the degree of R.

If f is not a polynomial but a set of polynomials then this procedure computes those subfields that the elements of f have in common. Each of these polynomials must be irreducible over K, otherwise this procedure may not work correctly.

$\mathrm{evala}\left(\mathrm{Subfields}\left(x^4+1\,2\right)\right)$

$\left\{\mathrm{RootOf}\left({\mathrm{\_Z}}^2-2\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\,\mathrm{RootOf}\left({\mathrm{\_Z}}^2+2\right)\right\}$

$\mathrm{evala}\left(\mathrm{Subfields}\left(x^4+1\,3\right)\right)$

$\varnothing$

$\mathrm{evala}\left(\mathrm{Subfields}\left(\left\{x^4+1\,x^4+2\right\}\,2\right)\right)$

$\left\{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+2\right)\right\}$