Determination of irreducible polynomials over a given field

[catcherintherye]

I am required to find all irreducible polynomials of the form xsquared + ax + b over the field F3, I have the 9 cases infront of me, i can see when something is reducible say xsquared is p(x)q(x) where p=x, q=x, but i have particular difficulty seeing when something is irreducible, e.g i know that xsqd + x + 2 is but i don't know how to show it, just as i do not know how xsqd + x +1 is irreducible over F2, although i can see how xsqd + 1 is since xsqd +1 = xsqd + 2x + 1 =(x+1)sqd, how do i know that a similar trick could not have been employed to make xsqd + x + 1 factorise?

 

[matt grime]

It's just a quadratic. If it is reducible then the factors are linear. So it is reducible if and only if one of 0,1,2 is a solution.

 

[tacman]

Finding irreducible polynomials over a given field can be a challenging task, but there are some techniques that can help us determine if a polynomial is irreducible. In your case, you are required to find all irreducible polynomials of the form xsquared + ax + b over the field F3.

One way to determine if a polynomial is irreducible is by using the Eisenstein's criterion. This criterion states that if a polynomial of the form ax^n + bx^(n-1) + ... + k has a prime number p that divides all coefficients except the leading coefficient a, and p^2 does not divide the constant term k, then the polynomial is irreducible.

In your example, xsquared + x + 2 has coefficients 1, 1, and 2, which are all divisible by 1 (a prime number). Also, 1^2 does not divide 2, so by Eisenstein's criterion, xsquared + x + 2 is irreducible over F3.

For xsquared + x + 1, we can try to use the same technique. However, in this case, all coefficients are divisible by 1 and 1^2 also divides 1. So, Eisenstein's criterion cannot be applied here.

Another way to determine irreducibility is by using the fact that a polynomial of degree 2 is irreducible if and only if it has no roots in the given field. In F3, the only possible roots are 0, 1, and 2. So, we can check if xsquared + x + 1 has any of these values as a root. If none of them is a root, then the polynomial is irreducible. In this case, none of the values is a root, so xsquared + x + 1 is also irreducible over F3.

Lastly, we can also try to factorize the polynomial and see if it can be written as a product of two polynomials of lower degree. For example, if we try to factorize xsquared + x + 1, we get (x+1)(x+1) which is not of the form xsquared + ax + b. This shows that xsquared + x + 1 cannot be factorized and is therefore irreducible over F3.

In conclusion, there are different methods that can be used to determine irreducibility of a polynomial over a given field. In your case, you can