# Fields

• March 3rd 2008, 09:39 AM
joanne_q
Fields
ok here's the 2nd one:
Deduce which of the following are fields:
(i) $\mathbb{F}_5(x)/(x^2 + 2x + 2)$
(ii) $\mathbb{F}_5(x)/(x^2 + 2)$
(iii) $\mathbb{F}_5(x)/(x^2 +3)$
(iv) $\mathbb{F}_5(x)/(x^2 + 4)$

Show that $\mathbb{F}_5(x)/(x^2 + 2)$ and $\mathbb{F}_5(x)/(x^2 + 3)$ are isomorphic as rings.

for part (i) to (iv), im guessing i have to draw out their additive and multiplicative tables and deduce that all polynomials have remainder inverses for them to be fields? is this correct? or are there any other suggestions? thnx :)
• March 3rd 2008, 10:09 AM
ThePerfectHacker
$F[x]/(p(x))$ is a field for non-constant polynomial if and only if p(x) is irreducible.