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    We know that F_2[x]/(x^2 + x + 1) is a field of cardinality 4; call it F_4.
    Find an irreducible quadratic f(y) [element of] F_4[y].
    What is the cardinality of the finite field F_4[y]/(f(y))?
    This field is isomorphic to the splitting field for x^2^n - x over F_2[x]. What is the appropriate n in this case?

    Show that x^4 + x^3 + x^2 + x + 1 [element of] F_2[x] is irreducible.
    Find the cardinality of the field F_2[x]/(x^4 + x^3 + x^2 + x + 1).
    This field is isomorphic to the splitting field for x^2^n - x over F_2[x]. What is the appropriate n in this case?

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    1. The elements of the finite field are 0, 1, , and , where An irreducible quadratic polynomial over is The cardinality of is is isomorphic to the splitting field of over , i.e. we have

    2. To see that the polynomial is irreducible, first we show that it has no roots in . Since we have we see that neither 0 nor 1 is a root. Next we must show that is not the product of two quadratic polynomials in . Suppose it were. Then we would have whence The last of these equations implies and implies that one of a and c is 0 and the other is 1, which in turn implies , whence the second equation becomes But since we have yielding a contradiction. Thus is not the product of two quadratic polynomials, so it is irreducible.

    The cardinality of is K is isomorphic to the splitting field of over , i.e. we have

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:21 pm ad1c9bdddf>
    https://brainmass.com/math/algebra/fields-cardinality-splitting-fields-467403

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