0

I'm trying to generalise results on extensions of field morphisms presented in texts on Galois theory. Is the following statement true ?

Suppose $K = k(\{\alpha_i\}_{i \in I})$ is a (not-necessarily finite) algebraic field extension and $L / k$ is a field extension such that $m_{\alpha_i, k}(X)$ splits completely in $L$. Then there exists a $k$-field morphism $\sigma : K \rightarrow L$.

Two particular cases of interest being if $L = \bar{k}$ or if $K,L$ are both splitting fields.

david r.
  • 81
  • Field morphisms are always inclusions so a first question might be if the $\alpha_i$ are in $L$? – Jeroen van der Meer Jan 28 '21 at 09:31
  • Yes. This is true. The argument is similar to the one I gave as an answer here. If the set $I$ were finite you would do this by induction on $|I|$. When that is not known (or known to be false) Zorn's lemma takes over the duties of an inductive step. – Jyrki Lahtonen Jan 28 '21 at 10:25
  • Not voting to close as a duplicate. At least not right away. That's because A) I have [tag:field-theory] dupehammer, so my vote would take effect immediately, B) the duplicate is not exact (there $\overline{k}$ was assumed, here we would use the assumption that the minimal polynomials split), C) I answered there, and I find forcing everyone interested in this question to read my answer is surely bad form. For all I know there may be a better duplicate target somewhere :-) So I want to leave a window of opportunity for opposite arguments before acting (in spite of the action being reversible!). – Jyrki Lahtonen Jan 28 '21 at 10:28
  • I don’t think this is a duplicate, rather a generalisation. It also shows uniqueness of splitting fields rather than just algebraic closure – david r. Jan 28 '21 at 11:24

0 Answers0