Polynomials

Warning A polynomial is not a function! A polynomial gives rise to a function, as we’ll recall in a moment. But a polynomial itself is a purely formal object. To emphasize this, we sometimes call the symbol 𝑡 an indeterminate rather than a ‘variable’.

We've already seen different examples of fields: the rational field ℚ, the field ℝ and ℂ as well as finite fields as ℤ_p, with p prime number. In the following treatment we shall indicate with K a generic field and demostrate the theorems valid for any field.

Definition 5.1.1 A Polynomial function or rational integer function over the field K is a function p from K to itself such that there exists n ≥ 0 elements a_i ∈ K so that

Let ℱ the set of all polynomial functions from K to itself. We can define addition and multiplication of such functions in K, by using the addition and multiplication operations defined in K. If p(x) and q(x) are the polynomial functions

The sum of the polynomials function p(x) and q(x) is defined to be

The product of the polynomials p(x) and q(x) is defined to be

We now pass from polynomial functions to polynomial and see the difference.

Definition 5.1.2 A polynomial p(x) with coefficients in a field K is a formal expression of the form

where x is a variable.  □

It follows from the definition that two polynomials p(x) = ∑ⁿ_j=0 a_ixⁱ and q(x) = ∑ⁿ_i=0 b_i xⁱ, with a_i, b_j ∈ K, are equal if and only if a_i = b_j, ∀i (in particular, if m > n, then b_{n + 1} = b_n+2 = ··· = b_m = 0.

The set of all polynomial with coefficients in K is denoted by K[x].

For example: Expressions, like √x + 3x + 9, are not a polynomial because all the powers of variable x are not whole numbers.

Polynomial vs Polynomial functions

Usually Polynomial and Polynomial functions are indicated in the same way as p(x) = a_n xⁿ, but there is a clear distiction between them. Two polymonial function are equal when they assume the same value for a given x ∈ K. On the other hand two polynomials are the same if they have the same formal expression (5.1.1). It is obvious that a polynomial p(x) = a₀ x⁰ + a₁x¹ + ... + a_nxⁿ defines a polynomial function which maps a c in K to p(c) = a₀ c⁰ + a₁c¹ + ... + a_ncⁿ also in K. But if we consider the mapping

such application is for sure surjective but can be non-injective i.e. it can happen that different polynomials give rise to the same polynomial function. For example if K = ℤ₃ the two polynomials f(x) = x² and g(x) = x³ + x² − x correspond to the same polynomial function from ℤ₃ to ℤ₃.

ℤ₃ = {[[0]₃,[1]₃, [2]₃} so if we take the element [2]₃ we have replacing into the two functions: f([2]₃) = [2]² = [4] and g([2]) = [2]³ + [2]² − [2] = [10 but [4]₃ and [10]₃ represent the first class.

For a finite field K, Ψ is never injective: K[x] is a infinite set because there are infinite polynomial expression we can write (even if coefficients are part of a finite set, the integer n can assume arbitrary values in ℕ). We'll soon show that when the field K is infinite Ψ is biunivoque thus it is possible in this case to make no distinction between polynomial and polynomial functions.

In what follows we'll focus our attention on polynomials even if is some cases it will be useful to consider them polynomial functions.

We can use for K[x] the same addition and multiplication defined for ℱ, with regards to these operations K[x] is a multiplicative ring with unit but there is a difference with ℱ: the zero polynomial is defined has the one with all coefficients a_i = 0. The opposite of the polynomial p(x) is the polynomial with all coefficients oppisite to a_i.

Definition 5.1.3. Let p(x) = ∑ⁿ_k=0 a_kx^k, then the degree of p(x) denoted by ∂p(x) or deg p is the largest integer m such that the coefficient of x^k is not zero.  □

The polynomials of degree one are called linear, those of degree two quadratic, those of degree three cubic, and so on.
Recall that the leading coefficient, a_n, of a polynomial p(x) is the coefficient of the highest power of x occurring in p(x); we say that a polynomial p(x) is monic if its leading coefficient is 1.
Notice that the degree of a non-zero constant p(x) = a₀ is zero. No degree is usually assigned to the zero polynomial, that is, the polynomial with all coefficients equal to zero.

Proposition 5.1.4. Let K be an integral domain or a field. Then the set K[x] forms an integral domain.

Proof. Let p(x) = ∑ⁿ_j=0 a_ixⁱ and q(x) = ∑^m_j=0 a_ixⁱ be non-zero polynomial (at least one coefficient is ≠ 0) of degree n and m which means that a_n ≠ 0 and b_m ≠ 0. From the product definition of two polynomials we have that the coefficient of x^n+m is a_nb_m because both a_n and b_n are different from zero and they are part of a field which does not contain zero-divisors. Thus the product p(x)q(x) cannot be equal to zero.□

Example 5.1.5. Let p(x) = 2x² + 4x + 2 and q(x) = 3x + 3 two polynomials of ℤ[x], we can verify thta

However if f(x) = p(x) = [2]x² + [4]x + [2] and g(x) = [3]x + [3] are polynomials in ℤ₆ we have

Let us explicitly remark the relations between the degrees of two polynomials with coefficients in a field, or in an integral domain, and the degrees of their sum and their product:

In the definition of a polynomial we can disregard the variable x as its value does not influece the definition. We can give the following definition of polynomial.

Definition 5.1.6. A polynomial with coefficients in K is an infinite sequence

of elements in K of which the a_i are zero from a certain index i onward.  □

We usually set a_i = 0 for each i > n; In the set of all these sequences we define addition and multiplication of sequences in the following way