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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:


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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Hi, I tried cut the second partial question in your other question, which is the same (I believe). – VVV Dec 22 '11 at 16:04
It is not clear to me, what is a normed polynomial? Does it mean monic = the leading coefficient must be equal to 1? Does it mean irreducible = cannot be written as a product of two linear polynomials. Usually irreducible polynomial is normalized to be monic. – Jyrki Lahtonen Dec 22 '11 at 16:04
Hi Jyrki, I think it is the same a monic. – Arthur from Belgium Dec 22 '11 at 16:11
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. – Jyrki Lahtonen Dec 22 '11 at 16:18
up vote 5 down vote accepted

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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Hi Matt, there was a faux-pas on my side: the question asks for all irreducible ones. If you mind, I will ask another question and leave this one as it is (and accept your answer). Sorry. – Arthur from Belgium Dec 22 '11 at 16:16
Hi @ArthurfromBelgium: No problem. I can delete my answer and you can edit your question, whichever you prefer. I think deleting my answer is the better solution. – Rudy the Reindeer Dec 22 '11 at 16:19
Let me know which you prefer @ArthurfromBelgium. – Rudy the Reindeer Dec 22 '11 at 16:22
Hi Matt, you shouldn't lose your points over my stupidity. I made a new one. – Arthur from Belgium Dec 22 '11 at 16:26
@ArthurfromBelgium I don't mind so much, don't worry about my points. I'm fine either way ; ) – Rudy the Reindeer Dec 22 '11 at 16:28

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