Algebraic element

Subalgebra generated by an algebraic element

Let be a K-monoid over a field and be an algebraic element with minimal polynomial . Then the unital subalgebra generated by is¹ falg

and is isomorphic to

Proof

Let First we will show ^eq1. First note that the RHS is clearly a vector subspace, so it suffices to show that for all . Applying the division algorithm for polynomials

where . But

so .

For the second statement, let . It follows from above that to every there corresponds a unique with such that . Let be the map

Now is a ring isomorphism, since for any

with an inverse by the evaluation map .

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Footnotes

1. Stated without proof in 2008. Advanced Linear Algebra, §18, p. 259. ↩