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Math Help - Galois Theory Help

  1. #1
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    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.

    Thanks in advance for your help.
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  2. #2
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    Quote Originally Posted by ShootTheBullet View Post
    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.

    Thanks in advance for your help.
    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.
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