1. ## Finite Fields

I need a little help im close but stuck.

1. Is there a field with 25 elements?

Well I know that |V_F|=p^n, where p is a prime. Then |V_F|=p^n=25=5^2, p=5, n=2. So now I have to construct the monic irreducible polynomial I know that it is degree 2 but how do you get the correct coefficients? I think the field is related to Z_5/p(x). but im stuck.

2. If W_F has 49 element, the what are the possibilities for F. Describe the collection of subspaces of W_F for each case.

thanks for the help.

2. Originally Posted by GreenandGold
1. Is there a field with 25 elements?

Well I know that |V_F|=p^n, where p is a prime. Then |V_F|=p^n=25=5^2, p=5, n=2. So now I have to construct the monic irreducible polynomial I know that it is degree 2 but how do you get the correct coefficients? I think the field is related to Z_5/p(x). but im stuck.
Remember what was said in your last post about this. Let $\displaystyle p(x) = x^2 - 2$. Check that this polynomial is irreducible and then construct $\displaystyle \mathbb{Z}_5[x]/(x^2 - 2)$. This would be a field with 25 elements.

2. If W_F has 49 element, the what are the possibilities for F. Describe the collection of subspaces of W_F for each case.
What do you mean by this problem?

3. How would I construct a polynomial in general for say degree n? degree 10 or something....

2. If V_F has 49 elements, the what are the possibilities for F. Describe the collection of subspaces of V_F for each case.

4. Originally Posted by GreenandGold
How would I construct a polynomial in general for say degree n? degree 10 or something....
There always exists an irreducible polynomial in general for a given degree over a finite field.
The problem is I do not know of any "nice" ways of finding them. What you are asking is basically, "is there an easy way to know if a number is prime or not?". There is no easy way to know if a number is prime and if you are made to search for primes you just use trail and error. The same with polynomial. You start listing polynomials of degree n and then hoping that you found an irreducible one. But the real problem is determining if a polynomial is irreducible or not. There is no efficient way to do this. So doing this problem by hand is a horrible. There are computer programs that can do it fast. You should find yourself a computer program if you want to actually start finding irreducible polynomials.

5. Are there any free programs? I dont want to invest in one. Should I just try this and see if it is reduce able over Z_p?

What about the other question? Are the only possiblities for F just 7 since p^n=7^2, 7 a prime.

6. Originally Posted by GreenandGold
Are there any free programs? I dont want to invest in one. Should I just try this and see if it is reduce able over Z_p?