By Kreuzer and Robbiano
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Example text
We can assume that gr = 1 and deg(gi ) > 0 for 1 ≤ i < r . 2) For all κ ∈ K , compute gcd(f, g1 − κ) and obtain a representation f = κ∈K gcd(f, g1 − κ) . If this representation contains r different non-constant factors, return it as the result. 3) For i = 1, 2, . . , let f = fi 1 · · · fi µi be the representation of f computed so far. For every κ ∈ K and every j ∈ {1, . . , µi } , compute gcd(fi j , gi+1 − κ). Then check, if the representation f= µi j=1 κ∈K gcd(fi j , gi+1 − κ) consists of r different non-constant factors.
Foundations Tutorial 4: Euclidean Domains In general, it is difficult to decide whether a given ring is factorial, and consequently there exists only a rather limited supply of examples of factorial domains. The purpose of this tutorial is to provide the reader with a tool for constructing or detecting a special kind of non-trivial factorial domains. We say that (R, ϕ) (or simply R ) is a Euclidean domain if R is a domain and ϕ is a function ϕ : R \ {0} → N such that for all a, b ∈ R \ {0} the following properties hold.
Which checks whether a given polynomial f ∈ K[x] is irreducible. Hint: You may use the CoCoA function Syz(. ) to compute the kernel of a linear map. f) Consider the following sequence of instructions. 1) Compute the matrix Q and the number r defined above. Let {(vi1 , . . , vid ) | 1 ≤ i ≤ r} be a K -basis of ker(Q − Id ) and gi = vi1 + vi2 x + · · · + vid xd−1 ∈ K[x] for 1 ≤ i ≤ r . g. we can assume that gr = 1 and deg(gi ) > 0 for 1 ≤ i < r . 2) For all κ ∈ K , compute gcd(f, g1 − κ) and obtain a representation f = κ∈K gcd(f, g1 − κ) .
Computational Commutative Algebra by Kreuzer and Robbiano
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