If I want to show that a certain polynomial is irreducible over a finite field, which methods do I have?
In particular how can I show that $X^4-3$ is irreducible over $\mathbb F_5$
The idea which I posted here does not work very well for me.
Its not hard to see that the polynomial has no roots in the finite field. So the only possibility being reducible is that I can split it into two polynomials with degree $2$.
Should I try it as usual with $x^4-3=x^4-2x^2+9=(a_1x^2+b_1x+c_1)(a_2x^2+b_1x+c_1)$?
I tried it, but didn't came to a solution. Can someone help me?