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futurebird
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Homework Statement
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 b can't satisfy 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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