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September 22nd, 2012, 10:53 AM   #1
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Finite field

Hello guys.

Can i ask you for some help about two problems i am fighting with?
The first one is:
Given is a polynom in . I have to show that is a finite field.

Is it enough if i show that the polynom is irreducible over ?

The second problem:
To prove that cannot be a field, no matter what the field is.

I would be very thankful if you could give me some hints how to solve these problems. Thank you in advance!
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September 22nd, 2012, 01:47 PM   #2
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Re: Finite field

for your first question: yes.

if p(x) is irreducible over F, then (p(x)), the ideal generated by p(x) is maximal.

for if I is an ideal with (p(x)) < I < F[x], then since F[x] is principal, we have I = (f(x)) with f(x)|p(x), contradicting the irreducibility of p(x),

or f(x) is a unit, in which case I = F[x] (the units of F[x] are the units of F), because I contains 1.

now, for any domain R, with I maximal in R, R/I is a field...because it has no non-trivial proper ideals.

for your second question: is there a typo?
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September 22nd, 2012, 02:26 PM   #3
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Re: Finite field

Deveno, thank you very much!

Yes, there is a mistake in the second problem. It should be
I'm sorry about that.
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September 22nd, 2012, 02:32 PM   #4
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Re: Finite field

That's important because now has an even number of terms and x= -1 is a root. Of course, since every field has a multiplicative identity, 1, every field contains -1.
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September 23rd, 2012, 12:54 PM   #5
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Re: Finite field

HallsofIvy, thank you very much!
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