Math Is Fun Forum

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Ricky, I need your help

Well, anyone who knows about Galois theory can help me here.

I’m trying to construct finite fields with

elements, where p is a prime and n > 1 is an integer. I’ve come up with

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i.e. the quotient ring of the ring of polynomials over the field of integers modulo p by the principal ideal generated by

, an irreducible polynomial of degree n. The ideal generated by the irreducible polynomial is maximal; since it is also a proper ideal, the quotient ring must be a field. A typical element in it has the form

,

; there are p possible values for

for each

and so there are

elements altogether. Hence it’s a finite field with

elements.

Now my question is: given any p and any n (> 1), can we always find an irreducible polynomial of degree n in

? Ricky? Anyone?

Last edited by JaneFairfax (2011-01-22 02:44:03)

David Bundy

Guest

Re: Ricky, I need your help

Yes, we know that the elements of the field of

elements are precisely the zeros of the polynomial

in

. Furthermore the elements of this field which are not in the field of

elements must be the zeros of irreducible polynomials of degree precisely n in

, since the degree of the Galois extension which they generate over

must be exactly n, otherwise they would be elements of the smaller field.

David Bundy

Guest

Re: Ricky, I need your help

Sorry that should be the polynomial

. There is also a little work required to show that all the zeros of the polynomial are distinct and form a field.