September 22nd, 2012, 10:53 AM  #1 
Newbie Joined: Sep 2012 Posts: 11 Thanks: 0  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! 
September 22nd, 2012, 01:47 PM  #2 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  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 nontrivial proper ideals. for your second question: is there a typo? 
September 22nd, 2012, 02:26 PM  #3 
Newbie Joined: Sep 2012 Posts: 11 Thanks: 0  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. 
September 22nd, 2012, 02:32 PM  #4 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  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.

September 23rd, 2012, 12:54 PM  #5 
Newbie Joined: Sep 2012 Posts: 11 Thanks: 0  Re: Finite field HallsofIvy, thank you very much!


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