Thread: Quotient ring and field question

1. Quotient ring and field question

For which prime numbers p is the quotient ring $F_5[x]/(x^2+1)$ a field?

From theorems in my notes, I know this quotient ring is a field if $(x^2+1)$ is a maximal ideal, i.e $(x^2+1)$ is irreducible modulo p over the field $F_5[x]$.

I don't see where to go from here though.. any help would be great
Thank you!

2. Originally Posted by gizmo
For which prime numbers p is the quotient ring $F_5[x]/(x^2+1)$ a field?

From theorems in my notes, I know this quotient ring is a field if $(x^2+1)$ is a maximal ideal, i.e $(x^2+1)$ is irreducible modulo p over the field $F_5[x]$.

I don't see where to go from here though.. any help would be great
Thank you!

Twice you wrote $\mathbb{F}_5[x]$ when it should, imo, be $\mathbb{F}_p[x]$

Hint: -1 is a quadratic residue modulo p iff $p=1\!\!\!\mod 4$

Tonio

Ps. By the way, precisely in $\mathbb{F}_5[x] \,,\,\,(x^2+1)$ is a not a maximal ideal.

3. oops sorry! yes I do mean $F_p[x]$

so, $x^2=-1modp$ iff $p=1mod4$

which is saying $(x^2+1)$ is reducible iff $p=1mod4$

but we want to find p when $(x^2+1)$ is irreducible:

$x^2 \neq -1modp$ or $p \neq 1mod4$

I'm not really sure what to do from here though :/

4. Originally Posted by gizmo
oops sorry! yes I do mean $F_p[x]$

so, $x^2=-1modp$ iff $p=1mod4$

which is saying $(x^2+1)$ is reducible iff $p=1mod4$

but we want to find p when $(x^2+1)$ is irreducible:

$x^2 \neq -1modp$ or $p \neq 1mod4$

I'm not really sure what to do from here though :/

You have it all above! First, a pol. of degree $\leq 3$ is reducible (over a given field) iff it has a root there; next, an ideal $ \leq \mathbb{F}[x]\,,\,\,\mathbb{F}$ a fiedl, is maximal iff $f(x)$ is irreducible.

So now you know exactly over what fields $\mathbb{F}_p$ the polynomila $x^2+1$ is irreducible , so...

Tonio

5. so if p $\neq$ mod4 ... p must satisfy one of these:

p=0mod4 but this is not true as a prime number can't be divided by 4.
p=2mod4 but this is also not true as it would mean p is even.

Therefore, p=3mod4, in order for the quotient ring to be a field.

I think that's it
Thank you for your help! I would rep you if I could again