symmetric monoidal (∞,1)-category of spectra
If a subset of a field is a subfield, then we call the larger field an extension of the smaller field .
More generally, if is any ring homomorphism between fields, then it must be an injection, so we may treat it as a field extension.
Let be a set, and let be a field. Then a field extension of is a commutative -algebra with injections such that for all , ensuring that is a field.
Every field extension can be factorized as a purely transcendental extension? followed by an algebraic extension. Indeed, by Zorn's lemma, we may construct a transcendence basis (i.e. maximal algebraically independent set) , and the purely transcendental part is the subfield generated by .
Unfortunately, this does not yield an orthogonal factorization system: given a field , we may form the field of rational functions over , which is a purely transcendental extension of , and we may form the algebraic closure , which is an algebraic extension of ; but we have the following commutative diagram,
where is the subfield of generated by , and is algebraic, yet there is no homomorphism making both evident triangles commute.
Last revised on May 8, 2021 at 03:53:57. See the history of this page for a list of all contributions to it.