Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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:

14.

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.

share|improve this question
    
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
2  
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
1  
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

1 Answer 1

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 $a_0 + a_1 x + a_2 x^2 + x^3$.

share|improve this answer
    
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. –  Matt N. Dec 22 '11 at 16:19
    
Let me know which you prefer @ArthurfromBelgium. –  Matt N. Dec 22 '11 at 16:22
1  
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
1  
@ArthurfromBelgium I don't mind so much, don't worry about my points. I'm fine either way ; ) –  Matt N. Dec 22 '11 at 16:28

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.