Changing the Coefficient Field

The standard constructions described in section 31.5 for \(R\)-modules may be applied to vector spaces. In addition, we may extend or restrict the field of scalars, using the functions described here.

ExtendField(V, L): ModTupFld, Fld -> ModTupFld, MapHom

Given a \(K\)-vector space \(V\), with \(K\) a field and \(L\) an extension of \(K\), construct the \(L\)-vector space \(U = V \otimes_K L\). The function returns

(a)

the vector space \(U\); and

(b)

the inclusion homomorphism \(\phi\ : V \rightarrow U\).

RestrictField(V, L): ModTupFld, Fld -> ModTupFld, MapHom

Given a \(K\)-vector space \(V\), with \(K\) a field and \(L\) a subfield of \(K\), construct the \(L\)-vector space \(U\) consisting of those vectors of \(V\) having all of their components lying in the subfield \(L\). The function returns

(a)

the vector space \(U\); and

(b)

the restriction homomorphism \(\phi\ : V \rightarrow U\).

VectorSpace(V, F): ModTupFld, Fld -> ModTupFld, Map

KSpace(V, F): ModTupFld, Fld -> ModTupFld, Map

KMatrixSpace(V, F): ModTupFld, Fld -> ModTupFld, Map

KModule(V, F): ModTupFld, Fld -> ModTupFld, Map

Given an \(n\)-dimensional \(K\)-vector space \(V\), and a subfield \(F\) of a finite field or cyclotomic field \(K\) such that \(K\) has degree \(m\) over \(F\), construct a vector space \(U\) of dimension \(mn\) over the field \(F\). The function returns

(a)

the vector space \(U\); and

(b)

a mapping \(\phi\ : V \rightarrow U\) such that a vector \((v_1, \ldots, v_i, \ldots, v_n)\) of \(V\) is mapped into the vector

\[(u_{11},\ldots, u_{1n}, \ldots, u_{i1}, \ldots, u_{in}, \ldots, u_{n1}, \ldots u_{nn} ),\]

where \((u_{i1}, \ldots, u_{in})\) is the field element \(v_i\) written as a vector over the subfield \(F\).