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Math Help - Doubt in Galois Field

  1. #1
    Junior Member classic_phohe's Avatar
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    Doubt in Galois Field

    1) Why do we have to specifically use only irreducible polynomials to
    construct fields? and are we obtain irreducible polynomial let say for the field of GF(2^2) ?

    2) Why primitive elements/polynomials are important?

    3) What does it means by conjugates. Here is a statement saying that
    neither /alpha nor its conjugates, /alpha^j are elements to which /beta is
    mapped.
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  2. #2
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    Quote Originally Posted by classic_phohe View Post
    1) Why do we have to specifically use only irreducible polynomials to
    construct fields? and are we obtain irreducible polynomial let say for the field of GF(2^2) ?
    Because if f(x) is not irreducible then \text{GF}(p)/f(x) is not necessarily a field!
    And so it cannot be an extension field.

    2) Why primitive elements/polynomials are important?
    Here is one example. Say K/F is primitive, then it means K=F(\alpha) for some \alpha \in K. If you want to compute \text{Gal}(K/F) all you need to do is compute the irreducible polynomial for \alpha over F and define the automorphism \alpha \to \alpha_i where \alpha_i is a root of this polynomial in K. Therefore, an easy way to see that K/F is Galois is if and only if K contains all the zeros of the minimal polynomial of \alpha with not repeated roots.

    3) What does it means by conjugates. Here is a statement saying that
    neither /alpha nor its conjugates, /alpha^j are elements to which /beta is
    mapped.
    Let K/F be an algebraic extension.
    Let \alpha,\beta \in K. We say that \alpha,\beta are conjugates iff they have the same minimal polynomials over F.
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