Factorisation of Polynomials

Question 1

What is meant by the content of the polynomial?

Answer 1

c(p(x)) is the h.c.f. of p(x).

Question 2

What is meant by primitive?

Answer 2

A polynomial p(x) in Z[x] is said to be primitive if p(x) ≠ 0 and the c(p(x)) = 1.

Question 3

If n in N and f in Z[x] then c(nf) = ?

Answer 3

n c(f)

Question 4

State and prove Gauss’s Lemma.

Answer 4

The product of two primitive polynomials f(x), g(x) in Z[x] is again primitive. i.e. if c(f) = c(g) = 1, then, c(fg) = 1.

Question 5

What’s the important corollary to Gauss’s Lemma, and prove it.

Answer 5

let f(x) in Z[x] be primitive. If f is reducible in Q[x] then f can be written as the product of two non-constant polynomials in Z[x].

Question 6

Is the converse to the Corollary to Gauss’s Lemma true?

Answer 6

Yes.
Let 0≠f(x) in Z[x] have deg(q(x)) ≥ 1. If q can be written as a product of two non-constant polynomials in Z[x], the f is reducible over Q because Z ≤ Q.

Question 7

Define µ_n, where n is a positive integer.

Answer 7

µ_n is the set of nth roots of unity 1.

Question 8

When is a complex number, w, primitive n-th root of unity?

Answer 8

when w^n = 1 but w^m ≠ 1, for all m=1,2,…,n-1.

or, if n=or(w).

Question 9

What is the n-th Cyclicotomic Polynomial?

Answer 9

ø_1 (x) = x-1 and ø_n (x) = ∏_((k,n)=1, 1≤k≤n-1) (x-e^(2πik/n) for n ≥ 2.

Question 10

define x^n - 1 in relevance of ø_n

Answer 10

ø_n (x) = (x^n - 1) / (∏_(l | n, l

Question 11

Let p be a prime number, then ø_p(x) = ?

Answer 11

ø_p (x) = x^(p-1) + x^(p-2) + … + x^2 + x ^ 1