1

I need to show the quotient $\frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle}$ is a field and find to which field is this quotient isommorphic.

I think this: $\frac{\mathbb{Z}[x]}{\langle 2x-1, 5 \rangle}\cong \frac{\mathbb{Z}_5[x]}{\langle 2x-1 \rangle}$. The polynomial $2x-1$ is irreducible in $\mathbb{Z}_5[x]$ thus, the quotient $\frac{\mathbb{Z}_5[x]}{\langle 2x-1 \rangle}$ must to be a field, but I don´t know to which is isomorphic.

Charly Moon
  • 11

1 Answers1

2

The ideal ${\langle 2x-1, 5 \rangle}$ is the kernel of the epimorphism from $\mathbb Z[x]$ to $\mathbb Z_5$ that sends $x$ to $3$ (and $1$ to $1$).

David
  • 19
  • 1
  • 1
    You probably mean "surjective morphism". Ring epimorphisms don't need to be surjective! See here: https://math.stackexchange.com/questions/81123 – user26857 May 09 '21 at 06:18