Homomorphisms

hom<F -> R | g>: FldFun, Rng, RngElt -> Map
hom<F -> R | cf, g>: FldFun, Rng, Map, RngElt -> Map: The homomorphism from the function field \(F\) to any ring \(R\) where \(g\) is the image of the generator of \(F\) in \(R\) and \(cf\) is a map from the coefficient field of \(F\) into \(R\).

hom< O -> R | g >: RngFunOrd, Rng, RngElt -> Map
hom< O -> R | cf, g >: RngFunOrd, Rng, Map, RngElt -> Map: Create the map from the order \(O\) of an algebraic function field to \(R\) using \(g\) as the image of the primitive element of \(O\). If the map \(cf\) is given it should be from the coefficient ring of \(O\) into \(R\), otherwise the coefficient ring of \(O\) should be automatically coercible into \(R\).

IsRingHomomorphism(m): Map -> BoolElt: Return whether the vector space homomorphism \(m\) is a homomorphism of rings.

Example: Homomorphisms of Relative Function Fields (ex-1107ee)

A simple use of homomorphisms is shown.

> PR<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(PR);
> FR1<a> := FunctionField(y^3 - x*y + 1);
> P<y> := PolynomialRing(FR1);
> FR2<c> := FunctionField(y^2 - a^5*x^3*y + 1);
> EFR2F := EquationOrderFinite(FR2);
> cf := hom<FR1 -> EFR2F | a + 1>;
> h := hom<FR2 -> EFR2F | cf, c + 1>;
> h(c) eq c + 1;
true
> h(a*c) eq a*c + a + c + 1;
true

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hom< O -> R | b₁, ..., bₙ >: RngFunOrd, Rng, RngElt, ..., RngElt -> Map
hom< O -> R | m, b₁, ..., bₙ >: RngFunOrd, Rng, Map, RngElt, ..., RngElt -> Map: Return the map from the order \(O\) of an algebraic function field into the ring \(R\) which maps the basis elements of \(O\) to \(b_1, ..., b_n\). The map \(m\), if given, should be from the coefficient ring of \(O\) into \(R\) and will be used to map the coefficients of the basis elements. If not given, the coefficient ring of \(O\) should automatically coerce into \(R\).

Homomorphisms#

Homomorphisms