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I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve:

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ii) If $\mathbf{F}$ is a finite field with $q$ elements, and $N_{d} = N_{d}(\mathbf{F})$ is the number of normed (monic), irreducible polynomials P in $\mathbf{F}[t]$ with degree d. What are $N_{2}$, $N_{3}$ ?

If there is anybody who understands these exercise, I would be very glad if you could explain them to me. Because when my professor tried it, I don't seem to have understood. Thank you for your attention.

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    It would be easier to give an answer, if you know about the Frobenius automorphism, minimal polynomials and such. It can be done without (following the hints in Gerry Myerson's answer), but the tally is a bit more cumbersome because of this - in particular for general $d$.2011-12-22

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If a polynomial of degree 2 is not irreducible, then it is a product of two linear polynomials. How many products of two linear polynomials are there?

If a polynomial of degree 3 is not irreducible, it is a product of three linear polynomials, or a linear and an irreducible quadratic. How many such things are there?

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    If this means that you passed up the opportunity to figure it out yourself from my suggestions, then I find my joy at your having solutions out$w$eighed by my sorrow at the missed chance.2011-12-23