Let $K$ be a subfield of $\mathbb{C}$ and let $f$ be an irreducible polynomial in $K$. Prove $f$ has no repeated roots in $\mathbb{C}$

Question

Let $K$ be a subfield of $\mathbb{C}$ and let $f$ be an irreducible polynomial in $K$. Prove $f$ has no repeated roots in $\mathbb{C}$.

Complete stuck, would I introduce an arbitary $\xi$?

Answer 1

In fact, every irreducible polynomial over a field of characteristic 0 has no repeated root over any extension of that field. However I will just consider the specific case you mentioned.

Assume that $f$ is an irreducible polynomial over $K$ and $\alpha\in \mathbb{C}$ is a root of $f$, then $f$ is a minimal polynomial of $\alpha$. If $\alpha$ is a multiple root of $f$ then $f'(\alpha)=0$. Hence $f\mid f'$, and it only happens when $f'=0$.

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However, an irreducible polynomial over a non-zero characteristic field can have a multiple root. For example, consider $f(x)=x^p-t^p$ over $\mathbb{F}_p(t^p)$. It is irreducible over $\mathbb{F}_p(t^p)$, but we can factor $f$ as $(x-t)^p$ over $\mathbb{F}_p(t)$. Note that $f'=0$ in this case as my previous proof suggests.