Factorisation in polynomial rings

Question 1

Reducible

Answer 1

Suppose deg(f(x))>0. We say f(x) is reducible if there exists h(x), k(x) e F[x] s.t. deg (h(x), deg(k(x) >0 and f(x) = h(x)k(x)

Question 2

Irreducible

Answer 2

If it is not reducible

Question 3

Highest common factor

Answer 3

Let f(x), g(x) e F[X]. The hcf of f(x) and g(x) is the polynomial h(x) e F[x] s.t.

a) h(x) is a monic polynomial
b) h(x) | f(x) and h(x)|g(x)
c) if k(x) e F[x], with k(x)|f(x) and k(x)|g(x) then k(x)|h(x)

Question 4

Primitive polynomial

Answer 4

If it is not imprimitive

Question 5

Imprimitive polynomial

Answer 5

Let f(X) = anx^n + … + a0 e Z[x]. We say that f(x) is imprimitive if there exists m e N such that m>1 and m | ai for all i.

Question 6

Product of primitive polynomials is primitive

Answer 6

Let f(x), g(x) e Z[x]. Suppose that f(x) and g(x) are primitive the f(x)g(x) is primitive.

Proof (by contradiction)

Question 7

Eisensteins Criterion

Answer 7

Let f(x) = anX^n + … + a0 e Z[X] and let p be prime. Suppose that

a) f(x) is primitive
b) p |/ an
c) p | ai for all i=0, 1, .., n-1
d) p^2 |/ ao

Then f(X) is irreducible