# Rings of Polynomials

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### Homework Problems

> :rocket: pg 207, 14.
> Find all zeros in the indicated finite field of the given polynomials with coefficients in the field. $x^3 +2x+2$ in $\mathbb{Z}_7$.

My approach was to try all the possible roots from 0~6 to see if there is any solution.

> :rocket: pg 218, 2.
> Find $q(x)$ and $r(x)$ as described by the division algorithm so that $f(x)=g(x)q(x) +r(x)$ with $r(x)=0$ or of degree less than the degree of $g(x)$.
> $f(x) = x^6+3x^5+4x^2-3x+2$ and $g(x) =3x^2+2x-3$ in $\mathbb{Z}_7 [x]$

Do division like normally but since we are in $Z_7$, we can adjust the coefficient to be in $Z_7$ during the division processes.

> :rocket: pg 218, 9
> The polynomial $x^4 + 4$ can be factored into linear factors in $\mathbb{Z}_5 [x]$. Find this factorization.

$$
x^4 + 4 \equiv x^4 - 1 \equiv (x^2+1)(x^2-1) \equiv (x+1)(x-1)(x^2-4) \equiv (x+1)(x-1)(x+2)(x-2)
$$

>:rocket: pg 219, 27.
Find all irreducible polynomials of the indicated degree in the given ring.
Degree 2 in $Z_2[x]$

All the possible polynomials in $\mathbb{Z}_2[x]$ are of the form $\{ax^2+bx+c | a,b,c \in \mathbb{Z}_2\}$. All **linear polynomials are irreducible**. For degree 2, we have $x^2,(x+1)(x+1) = x^2+1,x(x+1) = x^2+1$, all of which are reducible. Therefore, the only irreducible polynomial of degree 2 is $x^2 + x + 1$.

>:rocket: pg 219, 27.
>Find all prime ideals and all maximal ideals of $\mathbb{Z}_6.$

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