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Math Help - Construct a field

  1. #1
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    Construct 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?
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Construct a field of order 25

    I don't really understand this question, am I suppose to list the polynomials of degree 5?
    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 p(x) be a (nonconstant) polynomial over a field F. Then p(x) is irreducible over F if and only if F[x]/\left< p(x) \right> is a field.

    Consider F = \mathbb{Z}_5, and the polynomial p(x) = x^2 + 2 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 F[x]/\left< x^2+2 \right> is a field. How many elements are in this field? Note any element in this field can be written as ax+b since of uniquneness a and b can be any elements of the field F since there are five elements in total we can write 5\cdot 5 = 25.
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  3. #3
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    Thanks.

    So if I want to construct a field of order 27. I pick x^3 + 1 \in Z_{9}?
    Last edited by tttcomrader; November 25th 2007 at 06:23 AM.
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  4. #4
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    No, x^3+1 is reducible, you can write (x+1)(x^2+2x+1) or note that x=2 is a zero of this polynomial.
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