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This is a question from D.J.H. Garling's "A course in Galois Theory", and I am struggling to find an answer. Can anyone help me?

Suppose $K\leq L\leq L(\alpha)$ are field extensions, and that $|L(\alpha):L|$ and $|L:K|$ are relatively prime. Show that the minimal polynomial of $\alpha$ over $L$ has its coefficients in $K$.

Let $|L(\alpha):L| = n$ and $|L:K| = m$ where $(m, n) = 1$. I can prove that the minimal polynomial of $\alpha$ over $K$ must have degree which is a multiple of $n$. Hence, $|K(\alpha):K|=nt$ for some integer $t$. Let $|L(\alpha):K(\alpha)|=s$.

Now by the tower law, we have $nts = mn \implies ts =m$.

If I can show that $t=1$, then we are done. But why can't we have $ts=m$ where $t$ and $s$ are proper factors of $m$?

I know I need to use the fact that $(m,n) = 1$ somewhere, but I can't see where.

It's easy if we are given two quantities on 'different sides of the ladder' as coprime, but here $m$ and $n$ are on the same side of the ladder. If this is a typo in the book, is there a counterexample?

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  • $\begingroup$ Did you mean Galois extensions ? $\endgroup$
    – reuns
    Jan 5 '17 at 13:14
  • $\begingroup$ Not necessarily. Just finite extensions. $\endgroup$ Jan 5 '17 at 13:24
  • $\begingroup$ @Globe Theatre I know your post is already older, but I stumbled across the same problem in Garling's and in my version its $[K(a):K]$ and $[L:K]$ are coprime, not $[L( a):L$ and $[L:K]$ . Do you have any hints for this question? $\endgroup$
    – Marcel S
    Aug 25 '17 at 11:23
  • $\begingroup$ GlobeTheatre, I started a new post with the corrected version of Garling's question here: math.stackexchange.com/questions/2405708/… $\endgroup$
    – Marcel S
    Aug 25 '17 at 14:53
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Try $K=\mathbb{Q}$, $L=\mathbb{Q}(\sqrt{2})$, $\alpha=\sqrt[6]{2}$.

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Another counterexample to the exercise as written is to take $\alpha \in L \setminus K$. Then $|L(\alpha):L|=1$ is relatively prime to everything, but the minimal polynomial for $\alpha$ over $L$ is simply $x - \alpha$ which clearly does not have coefficients in $K$.

I think it is a typo and that Garling meant $|K(\alpha):K|$ and not $|L(\alpha):L|$. Another statement for the exercise, which is probably more to the point and has essentially the same proof, is:

Suppose that $K \subset L \subset L(\alpha)$ are field extensions, and that $|K(\alpha):K|$ and $|L:K|$ are relatively prime, then $|L(\alpha):L| = |K(\alpha):K|$.

A proof would be:

Since $|K(\alpha):K|$ and $|L:K|$ are relatively prime, and both divide $|L(\alpha):K|$, their product must divide $|L(\alpha):K|$, implying $|L(\alpha):K| \geq |K(\alpha):K| \cdot |L:K|$.

But then, the general fact that $|K(\alpha):K| \geq |L(\alpha):L|$ implies that $|K(\alpha):K| \cdot |L:K| \geq |L(\alpha):L| \cdot |L:K| = |L(\alpha):K|$.

So the inequalities collapse, forcing $|L(\alpha):L| = |K(\alpha):K|$.

The exercise in this form can be used, for example, in exercise 5.9 to show that the positive $p$-th roots of $2$, as $p$ varies over the primes, are linearly independent over $\mathbb Q$.

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