- #1

PsychonautQQ

- 784

- 10

## Homework Statement

f(x) = x^7 + 3x^6 + 3x^5 - x^3 - 3x^2 - 3x where the coefficients are elements of F_5. Show that this polynomial is divisible by x^5-x and construct a splitting field L for f over F_5 and computer [L:F_5]

## Homework Equations

## The Attempt at a Solution

So the first thing I did was turn all the negative coefficients positive so that f(x) = x^7 + 3x^6 + 3x^5 + 4x^3 + 2x^2 + 2x and we want to divide this polynomial by x^5+4x. Upon doing the long division, I get a quotient x^2 + 3x + 3 with remainder 4x^3... I checked my steps and I got the same answer with the same remainder, can anyone check my work here?