I want to get a better understanding of quotient rings so I have two questions.
Let $f(x) = x^2 + 2$
Let $R = \mathbb{Z}_{5}/(f(x))$
Now as $f$ is irreducible in $\mathbb{Z}_{5}$ we have that $R$ is a field with elements being all polynomials in $\mathbb{Z}_{5}$ with degree less than $2$, i.e.
$\{0, 1, 2, 3, 4,$
$x, x + 1, \dots, x + 4,$
$,\dots,$
$4x, 4x + 1, ..., 4x + 4 \}$
- First question - But how can, say, the element $x \in R$ be a unit?
Also now if we let the quotient be a reducible polynomial, the ring is not supposed to be a field? I.e.
Let $g(x) = x^2 + 1$
Let $S = \mathbb{Z}_{5}/(g(x))$
- Second question - It seems to me that $S$ will have exactly the same elements as $R$ and hence it will also be a field?