# Polynomial Rings: Irreducible Polynomials

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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

...

#### Solution Summary

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