Division algebra

Division algebra with only algebraic elements over an algebraically closed field

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

Proof

Let and be its minimal polynomial. Since has no zero divisors, must be an irreducible polynomial: 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

#state/tidy | #lang/en | #SemBr

Footnotes

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