How to construct $F_2[x]/(x^3+x^2+1)$ quotient ring and check whether it is a field?

Question

Could you please suggest me methods for constructing this kind of quotient rings and checking if they are field. For example, if we show that $(x^3+x^2+1)$ is irreducible, then does this imply that $F_2[x]/(x^3+x^2+1)$ is a field. Could you please help?

Answer 1

But $x^3+x^2+1$ is irreducible. Otherwise, it would be possible to express it as the product of two non-constant polynomials, one of which had to be of degree $1$ (since $x^3+x^2+1$ has degree $3$). But then it would have a root in $\mathbb{F}_2$, which is not the case.

Answer 2

Yes! If p(x) is irreducible then the ideal it generates will be a maximal ideal in F[x] where F is a field. And we have a well known result that R/I is a field iff I is maximal . hence the result

Hope it works

Answer 3

Fact 1: $\mathbb{F}_2[x]$ is a principal ideal domain, as the polynomial ring over a base field.

Fact 2: By the logic in José's answer, $x^3+x+1$ is irreducible.

Fact 3: The quotient of a ring by a maximal ideal gives a field.

These allow you to conclude.