–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)

–Hilbert’s 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 ⇒ Σu∈U u = 0

–∀f ∈ F[X] there is a splitting field E. And: F finite ⇒ E finite

–existence of primitive ...

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