Division algebra

Division algebra with only algebraic elements over an algebraically closed field

Let be an algebraically closed field and be a division algebra such that every is an algebraic element over .¹ Then . falg

Proof

Let and be its minimal polynomial. Since has no zero divisors, must be an ^irreducible: For if then and hence either or , a contradiction. Since is irreducible it is linear by ^A2, thus whence .

Corollaries

The following situations guarantee every element is algebraic over .

1. All elements of a finite-dimensional unital associative algebra are algebraic.
2. Dixmier’s lemma
3. Quillen’s lemma

---

tidy | en | sembr

Footnotes

1. Equivalently is an algebra such that every has a minimal polynomial with a nonzero constant term ↩