Polynomials over a field, Polynomial domain, Quotients of polynomials, Remainder theorem, Greatest common divisor, Unique Factorization Theorem

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Polynomial in λ over a field F. Let λ denote an abstract symbol (indeterminate) which is assumed to be commutative with itself and with the elements of a field F. The expression

(1) f(λ) = a_nλⁿ + a_n-1λ^n-1+ ... + a₁λ + a₀λ⁰

where the a_i’s are in the field F is called a polynomial in λ over the field F.

Zero polynomial. If every a_i = 0 in (1) above, it is called the zero polynomial and is written f(λ) = 0.

Leading term of a polynomial. The term of highest degree i.e. the term a_nλⁿ in (1) above.

Monic polynomial. A polynomial of the form

f(λ) = λⁿ + a_n-1λ^n-1+ ... + a₁λ + a₀λ⁰

where the coefficient of the leading term is 1.

Equal polynomials. Two polynomials which contain the same terms.

Polynomial domain F[λ] over a field F. The set of all polynomials in λ (of every degree: n = 0, 1, 2, ... , ∞) over the field F – the totality of all such polynomials.

The individual polynomials within the polynomial domain F[λ] can be added, subtracted and multiplied and the domain can be viewed as an abstract number system. Regarded as such it meets all the axiomatic requirements of a field except that it lacks a multiplicative identity element. It meets the axiomatic requirements of an integral domain. For example, the following laws hold:

f(λ) + g(λ) = g(λ) + f(λ)

( commutative over addition )

f(λ) ∙ g(λ) = g(λ) ∙ f(λ)

( commutative over multiplication )

If f(λ) 0 while f(λ) ∙ g(λ) = 0 , then g(λ) = 0

( no divisors of zero )

If g(λ) 0 and h(λ)∙ g(λ) = k(λ)∙ g(λ) , then h(λ) = k(λ)

( cancellation law holds )

Quotients of polynomials.

Theorem (Division algorithm). Let f(λ) and g(λ) ≠ 0 be polynomials in the polynomial domain F[λ] over the field F. Then there exist unique polynomials h(λ) and r(λ) in F[λ], where r(λ) is either the zero polynomial or is of degree less than that of g(λ), such that

(2) f(λ) = h(λ) ∙ g(λ) + r(λ)

Here r(λ) is called the remainder in the division of f(λ) by g(λ), If r(λ) = 0 , g(λ) is said to divide f(λ) and g(λ) and h(λ) are called factors of f(λ).

Note that the above theorem is exactly analogous to the division algorithm for integers: For any integer a and any positive integer b, there exist unique integers q and r such that

a = bq + r, 0 r < b

The integer a is the dividend, b is the divisor, q is the quotient and r is the remainder. [This theorem could be stated differently as “the quotient a/b equals q plus a remainder of r” – which explains the terminology.]

Let f(λ) = h(λ) ∙ g(λ). When g(λ) is of degree zero, that is, when g(λ) = c, a constant, the factorization is called trivial. A non-constant polynomial over F is called irreducible over F if its only factorization is trivial.

Irreducible polynomial. A polynomial that cannot be written as the product of two polynomials with degrees of at least 1 and having coefficients in some given domain or field. Unless otherwise stated, “irreducible” means irreducible in the field of the coefficients of the polynomial under consideration.

Remainder theorem. When a polynomial in x is divided by x - h, the remainder is equal to the number obtained by substituting h for x in the polynomial. More concisely, f(x) = (x-h)q(x) + f(h), where q(x) is the quotient and f(h) the remainder.

Example. If f(x) = x⁴ - 2x³ + 3x² - x + 2 is divided by x - 3, the quotient is q(x) = x³ + x² + 6x + 17 and the remainder is 53. If we substitute h = 3 into f(x) we get 3⁴ - 2 ∙ 3³ + 3 ∙ 3² - 3 + 2 = 53.

Proof. Let a polynomial f(x) be divided by x - h until a constant term is obtained. Then (2) above becomes

(3) f(x) = (x - h)g(x) + r

where g(x) is the quotient and r is a constant. If we substitute x = h into (3) we get

f(h) = (h - h)g(x) + r = 0 ∙ g(x) + r = r

Common divisor of two or more polynomials. A polynomial which is a factor of each of the polynomials. If h(x) divides both f(x) and g(x), it is called a common divisor of f(x) and g(x).

Greatest common divisor of two or more polynomials. A polynomial d(x) is called the greatest common divisor of polynomials f₁(x), f₂(x), ... ,f_n(x) if all the following hold:

● d(x) is monic

● d(x) is a common divisor of all the polynomials f₁(x), f₂(x), ... ,f_n(x)

● every common divisor of f₁(x), f₂(x), ... ,f_n(x) is a divisor of d(x)

Theorem 1. If f(x) and g(x) are polynomials in F[x], not both the zero polynomial, they have a unique greatest common divisor d(x) and there exist polynomials h(x) and k(x) in F[x] such that

d(x) = h(x) ∙ f(x) + k(x) ∙ g(x) .

This theorem has a counterpart in the realm of the integers: Any two integers a ≠ 0 and b ≠ 0 have a positive greatest common divisor (g.c.d.) which can be expressed as a “linear combination” of a and b in the form d = au + bv for integers u and v.

When the only common divisors of f(x) and g(x) are constants, their greatest common divisor is d(x) = 1.

Theorem 2. If the greatest common divisor of f(x) of degree n > 0 and g(x) of degree m > 0 is not 1, there exist non-zero polynomials a(x) of degree < m and b(x) of degree < n such that

a(x) ∙ f(x) + b(x) ∙ g(x) = 0

and conversely.

Relatively prime polynomials. Two polynomials are called relatively prime if their greatest common divisor is 1.

Theorem 3. If g(x) is irreducible in F[x] and f(x) is any polynomial of F[x], then either g(x) divides f(x) or g(x) is relatively prime to f(x).

Theorem 4. If g(x) is irreducible but divides f(x) ∙ h(x), it divides at least one of f(x) and h(x).

Theorem 5. If f(x) and g(x) are relatively prime and if each divides h(x), so also does f(x) ∙ g(x).

Unique Factorization Theorem. Every non-zero polynomial f(x) of F[x] can be written as

f(x) = c ∙ q₁(x)∙ q₂(x)∙ ... ∙ q_r(x)

where c ≠ 0 is a constant and the q_i(x) are monic irreducible polynomials of F[x].

References.

Ayres. Matrices (Schaum).