monic polynomials

In today's blog, I will present a property of monic polynomials. A monic polynomial is a polynomial of degree n where the coefficient of xⁿ is 1.

Lemma 1: Division by a monic polynomial

If f,g,h are polynomials such that f = g/h and g,h are monic. Then f is also monic.

Proof:

(1) Let g(x) = a₀x^r + a₁x^r-1 + ... + a_r-1x + a_r

(2) Let h(x) = b₀x^s + b₁x^s-1 + ... + b_s-1x + b_s

(3) Let f(x) = c₀x^t + c₁x^t-1 + ... + c_t-1x + c_t

(4) Since g(x) = h(x)*f(x), it follows that:

r = s + t

and

a₀ = b₀*c₀

(5) Since a₀ = b₀*c₀, it follows that:

c₀ = a₀/b₀

(5) Since g(x) is monic and h(x) is monic, it follows that a₀ = 1 and b₀= 1.

(6) It therefore follows that f(x) is monic since:

c₀ = a₀/b₀ = 1/1 = 1

QED

Roots of Polynomials

The following proof is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

Theorem: An element a ∈ F is a root of polynomial P ∈ F[X] if and only if (X-a) divides P.

Proof:

(1) deg(X - a) =1 [See Definition 4, here for definition of degree]

(2) Therefore, the remainder R of the division of P by (X - a) is a constant polynomial. [See Theorem, here]

(3) So, from the Division Algorithm for Polynomials (see Theorem, here), there exists Q,R such that:

P = (X - a)Q + R

(4) Further:

P(a) = (a -a)Q + R = R

(5) This shows that P(a) = 0 if and only if R = 0. That is, P(a) = 0 if and and only if P is divisible by (X - a).

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001