2

Here is another problem of the book Fields and Galois Theory by Patrick Morandi, page 37.

Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $K$ be a field extension of $F$ with $[K:F]=m$. If $\gcd(n,m)=1$, show that $f$ is irreducible over $K$.

I'm preparing for my midterm exam so I'm trying to solve as many as this book problems. Thanks for your helps.

Hoseyn Heydari
  • 1,547

1 Answers1

4

Let $L \supset K$ be obtained by adjoining a root of $f$ to $K$, say $\alpha$. Then consider the chains $F \subset K \subset L$ and $F \subset F(\alpha) \subset L$. You know $[F(\alpha):F] = n$, because $f$ is irreducible over $F$. What does this tell you about $[L:K]$?

Dustan Levenstein
  • 13,219
  • Are you reasoning for the absurd ?, are you assuming that $f$ is reducible over $K$ and so there exists such a $\alpha$? Or as? Why does this $\alpha$ exist? – Nash Oct 01 '17 at 18:14
  • 1
    @Nash Let $g$ be an irreducible factor of $f$, and let $L := K[x]/g(x)$. Then $L = K(\alpha)$ where $\alpha$ is the image of $x$ in $L$. – Dustan Levenstein Oct 01 '17 at 18:23
  • 1
    That's in answer to why $\alpha$ exists. As for what my reasoning is here, I'm just alluding to some basic divisibility theorems of chains of extensions. – Dustan Levenstein Oct 01 '17 at 18:24