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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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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