Irreducible polynomials mod 2

Number Theory Level 5

Let \( {\mathbb F}_2[x] \) be the ring of polynomials with coefficients in \( {\mathbb F}_2 = {\mathbb Z}/2{\mathbb Z} \). Recall that a polynomial is irreducible if it has no nonconstant factors of smaller degree.

For example, there are three irreducible polynomials of degree \( 4 \) in \( {\mathbb F}_2[x] \), namely \[ x^4+x+1, x^4+x^3+1, x^4+x^3+x^2+x+1. \]

How many irreducible polynomials of degree 17 are there in \( {\mathbb F}_2[x]?\)

Hint: read the finite fields wiki!

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