Algebra 3: algorithms in algebra [Lecture notes] - download pdf or read online

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Extra info for Algebra 3: algorithms in algebra [Lecture notes]

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To explain this, we denote by c(f ) the gcd of the coefficients of f ∈ Z [X], the so–called content of f . The first statement in the following proposition is usually called the Lemma of Gauss. 2 Proposition. a) The content is multiplicative in the sense that for nonzero polynomials f, g ∈ Z [X] the relation c(f g) = c(f ) c(g) holds. b) If f ∈ Z [X] factors as f = gh with g, h ∈ Q [X], then there exist a, b ∈ Q such that ag, bh ∈ Z [X] and f = (ag) (bh). Proof. a) The crucial case to consider is when c(f ) = c(g) = 1.

But this implies that we get a well-defined injective morphism em ), φ : Fq [X]/(f ) → Fq [X]/(f1e1 ) × · · · × Fq [X]/(fm e1 e m g + (f ) → (g + (f1 ), . . , gm + (fm )). The number of elements in Fq [X]/(f ) is q deg(f ) and the number of elements em ) is q e1 deg(f1 ) · · · q em deg(fm ) . Since these in Fq [X]/(f1e1 ) × · · · × Fq [X]/(fm numbers are equal the map φ is an isomorphism. This isomorphism (a polynomial version of the Chinese remainder theorem) enables us to translate statements about Fq [X]/(f ) into statements em ) and conversely.

This is the group of all bijections of the set {1, . . , n}. The binary operation is the composition of two bijections and the identity element is the identity map. 53 54 Algebraic prerequisites • The set of invertible n × n–matrices over a field K is a group with respect to matrix multiplication; the identity matrix serves as identity element. We denote this group by GL(n, K). 3 Definition. A group G is generated by g1 , . . , gn ∈ G if any g ∈ G can be written as a product of the gi ’s and gi−1 ’s.

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Algebra 3: algorithms in algebra [Lecture notes] by Hans Sterk


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