gcd of polynomials over a field

f(s) = 0 (X s) divides f(X)

R integral domain, f 0 f(X) R[X] has at most deg(f) roots

Let R be an integral domain, f 0. Then: root s of f is a simple root f (s) 0

F field, f F[X], f and f ' coprime all roots of f are simple

F field, f F[X]. Then: F[X]/f field f irreducible

F field, 0 f F[X] F[X]/f as additive group is isomorphic to Fdeg(f)

Hilberts basis theorem: R Noetherian and commutative R [X1, . . . , Xk] Noetherian

finite subgroups of the multiplicative group of a field are cyclic

the multiplicative groups of finite fields are cyclic

U finite subgroup of F, |U| 2 ΣuU u = 0

f F[X] there is a splitting field E. And: F finite E finite

existence of primitive ...

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