Computation in a ring and Galois Field

[ftneek]

I'm taking an applied abstract algebra course and am trying figure out how to check my work using the hp prime. Attached are a few problems we did in class. My professor gave me a skeptical look when I said I think my calculator is able to do these types of calculations, so I'd like to show them it can.

I worked out problem 4 by hand and got the inverse is x^3 + x + 1. On my calculator I entered GF(2,k^5 + k^3 + 1) and was able to enter inv(k^3 + 1) to obtain k^3 + k + 1, and the product is indeed 1.

Now when I try problem 1 I know there is an inverse because the gcd of the the two polynomials is 1 (in mod 2). But when I clear my variables and tried entering GF(2,k^4) I'm met with the error that k^4 is not irreducible. I was able to check my answer I got by hand by using the expression ((x^3 + x^2 + 1)%%2)*((x^3 + x^2 + 1)%%2) mod ((x^4)%%2). To find the inverse using my calculator I tried a few expressions such as inv(((x^3 + x^2 + 1)%%2) mod ((x^4)%%2)) but it returned a fraction which isn't what I was expecting. My question is how can I actually find this inverse using my calculator?

I'm still learning about Galois fields, I'm more familiar working in integers mod p or m. I think I understand it wouldn't be a field if the polynomial is reducible, so it makes sense why GF is giving that error. But is there some other way to perform these types of computations on the hp prime where the modulus polynomial is reducible?