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Non–algebraic) over K(θ1 , . . , θn−1 ). If there is no condition on being trancendental, then we simply speak of an elementary extension of K. Usually, K will be Q (x). 2 Remark. Note that if α is transcendental over K, then K[α] ∼ = K[X] (the ∼ polynomial ring in one variable over K) and K(α) = K(X) under the map sending X to α. This enables us to speak of the degree in α of a polynomial expression in α. 3 Theorem. (Liouville’s principle) Suppose the integral of f ∈ K exists in an elementary extension L of K, which has the same field of constants as K.

In that case multiply g and h by rational numbers a and b, respectively, in such a way that ag and bh have integer coefficients and content 1. Then c(abf ) = c(ag)c(bh) = 1 by a), so that ab = ±1. Altering the sign of a if necessary, we can arrange it so that ab = 1 and f = (ag) (bh). 3 Now an obvious operation on polynomials with integer coefficients is to reduce their coefficients modulo a prime. Let p be a prime and let f = am X m +am−1 X m−1 +· · ·+a1 X +a0 be a polynomial with integer coefficients.

Let G and H be two groups. A group homomorphism φ : G → H is a function satisfying φ(ab) = φ(a)φ(b) for all a, b ∈ G If φ is injective, φ is said to be a monomorphism. If φ is surjective, φ is said to be a epimorphism. If φ is bijective, φ is said to be an isomorphism. 1 (Rings) In these notes rings are commutative with unit 1. A ring is called a domain (or integral domain) if ab = 0 implies a = 0 or b = 0 for all a, b ∈ R. An element a ∈ R is called a unit (or invertible element) if there exists an element b ∈ R with ab = 1.

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