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I've trying to prove some theorem about Galois-Hopf Theory. It's details are not necessary to understand my question. I want to prove the following:

Let $F/K$ be a Galois field extension. Let $A$ be a $K$-algebra such that $A\subseteq F$. Then $A$ is a field.

I clearly see that $A$ must be a field, but I'm not sure how to prove it. I'm not sure if I can use minimal polynomials of elements of $A$ to find the expression of the inverse of any element in $A$, because that may not be defined unless $A$ is a field, which I haven't proved yet. That sounds like some kind of circular reasoning, so I considered posting it here. ¿Is this statement correct? ¿How can I prove it? Any help or hint will be appreciated, thanks in advance.

Alejandro Bergasa Alonso
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  • Minimal polynomials exist whether $A$ is a field or not: you take them over $K$, not over $A$. They exist because if $a\in A$ then $a\in F$, so you can look at the minimal polynomial over $K$ of the element $a$ viewed as an element of $F$. – Arturo Magidin May 28 '22 at 13:55
  • In fact, this argument works for any algebraic extension $F/K$! On the other hand, once some $x\in F$ is transcendental, then $K[x]\subset F$ is not a field. – Kenta S May 28 '22 at 14:05

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