The existence of polynomials in title has been asked as a problem on MathStack many times; some answers were using existence of finite bigger fields, and some answers concern Mobius function with counting argument. I didn't quickly stick to Mobius functions; so I was searching for some other proof of existence if there is.
I was thinking in the following way. To get irreducible polynomial of degree $n$, consider $n\times n$ matrices over the finite field $F$. Can't we find a special matrix whose characteristic polynomial is irreducible over $F$?