By Hans Sterk

**Read Online or Download Algebra 3: algorithms in algebra [Lecture notes] PDF**

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**Additional resources for Algebra 3: algorithms in algebra [Lecture notes]**

**Example text**

Before we state this and give the proof, we first explain an ingredient em . Consider the map of the proof. Suppose f factors as f = f1e1 · · · fm em ) φ : Fq [X] → Fq [X]/(f1e1 ) × · · · × Fq [X]/(fm e1 e g → (g + (f1 ), . . , gm + (fmm )) It is easy to verify that this map is a morphism of rings. The kernel consists e of the polynomials h ∈ Fq [X] such that fj j divides h for j = 1, . . , m. Since e the fj j are relatively prime, we conclude that h is in the kernel if and only if f | h. 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 ), .

G n } where e denotes the identity element. We can also write G = g . 6 Definition. Let G and H be two groups. If φ is surjective, φ is said to be a epimorphism. If φ is bijective, φ is said to be a isomorphism. 7 Example. Consider the group homomorphism φ : Sn → GL(n, K) defined by: to each element σ ∈ Sn we associate the matrix φ(σ) = (aij )ni,j=1 with aij = 1 if σ(j) = i, 0 otherwise It is easy to see that Sn and φ(Sn ) are isomorphic. Using this representation of Sn we can say that Sn is a subset of GL(n, K).

Then Φp (X) = Replace X by X + 1 and we find Φp (X + 1) = Xp − 1 . X −1 (X + 1)p − 1 . X The right–hand side works out as X p−1 + p p X + p. X p−2 + · · · + 2 p−1 This polynomial is suitable for the application of Eisenstein’s criterion for the prime p. We conclude that Φp (X) is irreducible. Φp (X) is part of a family of polynomials, the cyclotomic polynomials Φm (X) for m ∈ Z, m > 0. These are the minimal polynomials of e 2πi m and of fundamental importance in number theory. 1 Factoring a polynomial in Fq [X] (with q a power of the prime p) is a finite job.