Algebra Homomorphism - Examples

Examples

Let A = K be the set of all polynomials over a field K and B be the set of all polynomial functions over K. Both A and B are algebras over K given by the standard multiplication and addition of polynomials and functions, respectively. We can map each in A to in B by the rule . A routine check shows that the mapping is a homomorphism of the algebras A and B. This homomorphism is an isomorphism if and only if K is an infinite field.

Proof. If K is a finite field then let

p is a nonzero polynomial in K, however for all t in K, so is the zero function and our homomorphism is not an isomorphism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite).

If K is infinite then choose a polynomial f such that . We want to show this implies that . Let and let be n + 1 distinct elements of K. Then for and by Lagrange interpolation we have . Hence the mapping is injective. Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of A and B.

If A is a subalgebra of B, then for every invertible b in B the function which takes every a in A to b−1 a b is an algebra homomorphism (in case, this is called an inner automorphism of B). If A is also simple and B is a central simple algebra, then every homomorphism from A to B is given in this way by some b in B; this is the Skolem-Noether theorem.