91Ó°ÊÓ

Show that the Gaussian integer \(a+b i(b \neq 0)\) is prime if and only if the \(\operatorname{norm} a^{2}+b^{2}\) is an ordinary prime.

Use Gauss's method to solve the cyclotomic equation \(x^{6}+\) \(x^{5}+x^{4}+x^{3}+x^{2}+x+1=0\)

Show that the Galois group of the equation \(x^{5}-2\) over the rational numbers can be expressed as the group of substitutions of the form \(x^{\prime} \equiv a x+b(\bmod 5)\) and therefore has 20 elements.

Find a fifth-degree polynomial that is not solvable by radicals.

Prove that if the product of two matrices is the zero matrix, then at least one of the factors has determinant 0 .

Show that two equivalent quadratic forms have the same discriminant.

Create a field of order \(5^{3}\) by finding a third-degree irreducible congruence modulo 5 .