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Polynomial Rings: Irreducible Polynomials

Prove that x^2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x]/(x^2 + 1) is a field having 121 elements.

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Modern Algebra
Ring Theory (XXIX)
Polynomial Ring
Irreducible Polynomial

By:- Thokchom Sarojkumar Sinha

Prove that is irreducible over the field of integers mod
and prove directly that is a field having 121 elements.

Solution:- Here .
The field is .
Let , the ring of polynomials having coefficients in .
is a factor of if and only if for some .

Here
Hence

Therefore

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Solution Summary

Irreducible polynomials are investigated in this solution, which is detailed and well presented.

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