Irreducible Polynomials p 5 degree 4

In summary, in order to construct a field containing 625 elements in the form of Zn[x] mod f(x), where n is a prime number and f(x) is an irreducible polynomial of degree 4, one can find irreducible polynomials by considering the field F_5, which has 5 elements, and determining if there is a polynomial that all elements of F_5 satisfy. Precomputing such polynomials is not necessary.
  • #1
grandnexus
8
0
I am attempting to construct a field containing 625 elements and should be in the form Zn[x] mod f(x).

Factoring 625 leads to 5^4. So I'm guessing my field will be GF(5^4). So in order for me to construct a field with all elements in it, I need f(x) to be some irreducible polynomial mod 5 of degree 4.

How can I go about finding irreducible polynomials? I know I can choose all the polynomials below degree 4 with coefficients mod 5 and attempt to find one without factors, but that would take forever. Is there a quick way to do this or a list of precomputed irreducible polynomials given GF(p^n) where p is prime and n is greater than 1??

Thanks.
 
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  • #2
I can't imagine anyone's bothered to precompute anything. The solution is to think about F_5, the field with 5 elements, and decide if you know any polynomial that every element of F_5 satisfies.
 
  • #3


Constructing a field with 625 elements using the form Zn[x] mod f(x) is a common approach in abstract algebra. In order to find an irreducible polynomial of degree 4 mod 5, there are a few methods you can try.

One approach is to use the Eisenstein's criterion. This states that if a polynomial has the form x^n + a for some integer a, and a prime number p divides all the coefficients except the leading coefficient, then the polynomial is irreducible. In your case, you can try to find a polynomial of the form x^4 + a mod 5, and check if it satisfies the Eisenstein's criterion.

Another approach is to use the Berlekamp's algorithm. This algorithm is a systematic method for testing irreducibility of polynomials over finite fields. It involves finding a factorization of the polynomial over a larger field and then checking if the factors are also irreducible over the original field. This method may be more efficient than simply trying all possible polynomials of degree 4 with coefficients mod 5.

There are also precomputed lists of irreducible polynomials for certain fields, such as GF(p^n). These lists can be found online or in algebra textbooks. However, keep in mind that these lists may not cover all possible irreducible polynomials, so it's always best to double check with other methods.

I hope this helps you in finding an irreducible polynomial for your field with 625 elements. Good luck!
 

1. What is an irreducible polynomial of degree 4?

An irreducible polynomial of degree 4 is a polynomial of the form ax^4 + bx^3 + cx^2 + dx + e, where a, b, c, d, and e are coefficients and a is not equal to 0. This polynomial cannot be factored into polynomials of smaller degree with coefficients in the same field. It is considered the "building block" of all polynomials of degree 4.

2. How can I determine if a polynomial is irreducible of degree 4?

To determine if a polynomial is irreducible of degree 4, you can use the rational root theorem, which states that if a polynomial with integer coefficients has a rational root, it must be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. If there are no rational roots, the polynomial is irreducible.

3. What is the significance of irreducible polynomials of degree 4?

Irreducible polynomials of degree 4 have significance in many areas of mathematics, including algebra, number theory, and cryptography. They are used to construct finite fields, which have applications in coding theory and error-correcting codes.

4. Can an irreducible polynomial of degree 4 have complex coefficients?

Yes, an irreducible polynomial of degree 4 can have complex coefficients. In fact, it is often necessary to use complex coefficients when working with higher degree polynomials, as real coefficients may not be sufficient.

5. Are there any known techniques for factoring irreducible polynomials of degree 4?

Yes, there are techniques for factoring irreducible polynomials of degree 4, but they are not guaranteed to work for all polynomials. Some techniques include using the quadratic formula, factoring by grouping, and using the cubic formula. However, for many irreducible polynomials of degree 4, factoring may not be possible.

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