(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that any field of characteristic 0 is perfect.

2. The attempt at a solution

Let F be a field of characteristic 0.

Let K be a finite extension of F.

Let b be an element in K .

I need to show that b satisfies a polynomial over F having no multiple roots.

If f(x) is irreducible in F[x] then f(x) has no multiple roots.

I need to show that b satisfies a irreducible polynomial in F[x].

Well, suppose bcan'tsatisfy any irreducible polynomial in F[x]. Can I get a contradiction? What kind of element could I have that didn't satisfy any irreducible polynomial?

Then how can b be in the finite extension...? A finite extension for a field of characteristic 0 is of the form F(a), it is generated by a single element.

I'm stuck. I don't even know if what I've laid out so far is correct.

I'm having trouble connecting the arbitrary element b to a polynomial-- It's not obvious to me that b is the root of any polynomial in F[x].

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# Homework Help: Field of characteristic 0

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