Homework SoursePlease Show workSolutionZ5 0 1 2 3 4 is a field of order 5
Please Show work.
Solution
Z5 = {0, 1, 2, 3, 4} is a field of order 5. To find irreducible polynomials over Z5 of degree 2, we list all possible monic polynomials of degree 2 over Z5: x2 , x2 + 1, x2 + 2, x2 + 3, x2 + 4, x2 + x, x2 + 2x, x2 + 3x, x2 + 4x, x2 + x + 1, x2 + x + 2, x2 + x + 3, x2 + x + 4, x2 + 2x + 1, x2+2x+2, x2+2x+3, x2+2x+4, x2+3x+1, x2+3x+2, x2+3x+3, x2 + 3x + 4, x2 + 4x + 1, x2 + 4x + 2, x2 + 4x + 3, x2 + 4x + 4.
Every polynomial without a constant term has root 0, x 2 + 4, x2 + x + 3, x2 + 2x + 2 and x2 + 3x + 1 have root 1, x2 + 1 ,x2 + 3x, x2 + x + 4, x2 + 2x + 2 and x2 + 4x + 3 have root 2, x2 + 1, x2 + 2x, x2 + x + 3, x2 + 3x + 2 and x2 + 4x + 4 have root 3, and x2 + 4, x2 + x, x2 + 2x + 1, x 2 + 3x + 2 and x2 + 4x + 3 have root 4. So we will consider the remaining polynomials: x2 + 2, x2+ 3, x2 + x + 1, x2 + x + 2, x2 + 2x + 3, x2 + 2x + 4, x2 + 3x + 3, x2 + 3x + 4, x2 + 4x + 1 and x2 + 4x + 2 as monic irreducible polynomials of degree 2 in Z5 .
