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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) 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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    In that case, Matt has answered the question. The problem would be a bit more interesting, if it were about irreducible polynomials, but then you might need to know a bit more field theory.2011-12-22

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You need to count the number of polynomials of degrees $2$ and $3$ respectively.

A normed polynomial of degree $2$ looks like this: $a_0 + a_1 x + x^2$. In how many ways can you choose $a_0$ and $a_1$? You have $q$ possibilities to choose $a_0$ and $q$ possibilities to choose $a_1$. How many polynomials do you get?

Now you do the same for polynomials of the form $a_0 + a_1 x + a_2 x^2 + x^3$.

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    @ArthurfromBelgium I don't mind so much, don't worry about my points. I'm fine either way ; )2011-12-22