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How many monic irreducible polynomials of degree 6 are there in $F_5[X]$

Question: How many monic irreducible polynomials of degree 6 are there in F_5[X]?

Clearly such polynomials would be of the form x^6+ax^5+bx^4+cx^3+dx^2+ex+f=0 with the coefficients being 0,1,2,3,4 but am not sure how to proceed
(edited 8 years ago)
Original post by number23
Question: How many monic irreducible polynomials of degree 6 are there in F_5[X]?

Clearly such polynomials would be of the form x^6+ax^5+bx^4+cx^3+dx^2+ex+f=0 with the coefficients being 0,1,2,3,4 but am not sure how to proceed


There is a simple formula for the number of monic irreducible polynomials of degree d in Fp[X]\mathbb{F}_p[X]. It's called Gauss's formula and arises from a consideration of how xpn−xx^{p^{n}} - x factors into irreducibles. You can find this in Ireland and Rosen's "Classical Introduction to Modern Number Theory" page 84 (or google "how many monic irreducible"!)

Or are you expected to do this from first principles?
(edited 8 years ago)

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