How to find deg 4 and deg 5 irreducible polynomials over $\mathbb{F}_3$? 
Give examples of irreducible polynomials $f(x)$ and $g(x)$ in $\mathbb{F}_3[x]$ such that $\deg f(x)=4$ and $\deg g(x)=5$.

I can find irreducible polynomials of degree $2$ or $3$ by taking such polynomials which has no zeros in $\mathbb{F}_3$. But for $\deg 4$ and $\deg 5$ I don't know how to find a polynomial and then show that it will be irreducible.
Please suggest me any tips.
Thanks.
 A: A polynomial of degree $4$ or $5$ is irreducible if it has no linear or quadratic factor. There aren't many linear and quadratic polynomials over $\Bbb{F}_3$, so list them all and then just pick polynomials of degrees $4$ and $5$ that isn't divisible by any of them.
Of course you only need to consider monic polynomials. There are precisely $3$ linear monic polynomials over $\Bbb{F}_3$, and $9$ quadratic monic polynomials over $\Bbb{F}_3$, and only $3$ of them are irreducible. So there are only $9$ irreducible factors to avoid.

Alternatively you could take a more abstract approach, which may seem like magic if you aren't already very familiar with this topic:
Because $\Bbb{F}_3$ contains no primitive fifth root of unity, the fifth cyclotomic polynomial
$$\Phi_5=x^4+x^3+x^2+x+1$$
is irreducible over $\Bbb{F}_3$.
A similar trick works for degree $5$; there is of course no cyclotomic polynomial of degree $5$, but because $\Bbb{F}_3$ has no primitive eleventh root of unity, the eleventh cyclotomic polynomial $\Phi_{11}$ is irreducible over $\Bbb{F}_3$. Then for any root $\zeta$ of $\Phi_{11}$ the minimal polynomial of $\zeta+\zeta^{-1}$ is irreducible of degree $5$, and the minimial polynomial is
$$x^5+x^4-4x^3-3x^2+3x+1.$$
