degree of the complete field over a field

Asked Apr 16 '22 at 04:46

Active Apr 16 '22 at 04:46

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Consider some field $F$ and its algebraic closure $\bar{F}$. What are the possible values of $n_F = [\bar{F}:F]$? For example, if $F=\mathbb{Q}$, then $n_F=\infty$, however $n_\mathbb{R}=2$.

I'd like to know in general what values $n$ can take, that is, I want to find the set of all naturals $k$ for which there exists a field $F$ st. $n_F=k$. I have examples for $k=1,2,\infty$ however I don't have any others and would like to know if there are any examples.

asked Apr 16 '22 at 04:46

Mathew

1,894

Pretty sure this is a duplicate but no other values of $k$ are possible. – Eric Wofsey Apr 16 '22 at 05:41
If its a duplicate, do you know where the original question is? Thanks – Mathew Apr 17 '22 at 02:14
Almost, I would like to find a proof that these are the only possible values of k, however this is not given in either the wikipedia article or the MSE question. Where might I find such a proof? Thanks – Mathew Apr 18 '22 at 04:01
There's a proof at https://kconrad.math.uconn.edu/blurbs/galoistheory/artinschreier.pdf. – Eric Wofsey Apr 18 '22 at 04:20
Thanks, this is exactly what I wanted. I will close the question. – Mathew Apr 19 '22 at 18:58

degree of the complete field over a field

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