Construct a field of order 25.

I don't really understand this question, am I suppose to list the polynomials of degree 5?

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- Nov 24th 2007, 02:04 PMtttcomraderConstruct a field
Construct a field of order 25.

I don't really understand this question, am I suppose to list the polynomials of degree 5? - Nov 24th 2007, 02:28 PMThePerfectHacker
The question is saying to explicity create a field, we know a field of 25 elements exists but we want to explicitly state it.

I have to ways to to construct such a field I will do it with polynomials since that is what you mentioned and I assume that is what you want.

First we need to know a theorem: Let be a (nonconstant) polynomial over a field . Then is irreducible over if and only if is a field.

Consider , and the polynomial is irreducible since it is of degree two and it has no zero (just check for zeros x=0,1,2,3,4).

Now since it is irreducible by the theorem is a field. How many elements are in this field? Note any element in this field can be written as since of uniquneness and can be any elements of the field since there are five elements in total we can write . - Nov 25th 2007, 06:04 AMtttcomrader
Thanks.

So if I want to construct a field of order 27. I pick ? - Nov 25th 2007, 06:44 AMThePerfectHacker
No, is reducible, you can write or note that is a zero of this polynomial.