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    Prove that the polynomial x^2 + x + 1 is irreducible over

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    Prove that the polynomial x^2 + x + 1 is irreducible over F , the field of integers modulo 2.

    The complete problem is in the attached file.

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    https://brainmass.com/math/ring-theory/prove-polynomial-irreducible-over-619265

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    The solution is in the attached file.

    Algebra (II)
    By:- Thokchom Sarojkumar Sinha

    Prove that the polynomial
    is irreducible over F, the field of integers modulo 2.

    Solution:- Let

    Also let F be the field of integers modulo 2,
    then F = { 0,1} where the operations of F are addition modulo 2
    and multiplication modulo 2
    or,

    We have ,
    where

    Therefore

    ...

    Solution Summary

    It proves that the polynomial x^2 + x + 1 is irreducible over F , the field of integers modulo 2.
    It also defines the two operations addition modulo 2 and multiplication modulo 2 of the field of
    integers F.
    The solution is given in detail.

    $2.19

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