1. ## Galois Theory Help

Hi, I have an examination for Galois theory coming up shortly, and realised there's a question in one of our past papers I don't know how to answer, it reads as follows:

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

It's worth a fair few marks so I expect the answer's fairly long, but if anyone can provide any ideas how to deal with that I'd be very greatful.

2. Originally Posted by ShootTheBullet
Hi, I have an examination for Galois theory coming up shortly, and realised there's a question in one of our past papers I don't know how to answer, it reads as follows:

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

It's worth a fair few marks so I expect the answer's fairly long, but if anyone can provide any ideas how to deal with that I'd be very greatful.

if $a \in \mathbb{C}$ is a repeated root of $f,$ then $f(a)=f'(a)=0.$ since $f$ is irreducible over $K,$ we have $\gcd(f,f')=1,$ i.e. $u(t)f(t)+v(t)f'(t)=1,$ for some $u,v \in K[t] \subseteq \mathbb{C}[t].$ let $t=a$ to get $0=1. \ \Box$
note that since $\text{char} \ \mathbb{C} = 0,$ the derivative of a non-constant polynomial is not identically 0.