2.6. Polynomials

Unless otherwise stated, in this section we denote by K an arbitrary field and by K[X] the ring of polynomials in one indeterminate X and with coefficients from K. Since K[X] is a PID, it enjoys many properties similar to those of . To start with, we take a look at these properties. Then we introduce the concept of algebraic elements and discuss how irreducible polynomials can be used to construct (algebraic) extensions of fields. When no confusions are likely, we denote a polynomial by f only.

2.6.1. Elementary Properties

Since K[X] is a PID and hence a UFD, every polynomial in K[X] can be written essentially uniquely as a product of prime polynomials. Conventionally prime polynomials are more commonly referred to as irreducible polynomials. Similar to the case of the ring K[X] contains an infinite number of irreducible elements, for if K is infinite, then is an infinite set of irreducible polynomials of K[X], and if K is finite, then as we will see later, there is an irreducible polynomial of degree d in K[X] for every .

It is important to note here that the concept of irreducibility of a polynomial is very much dependent on the field K. If K ⊆ L is a field extension, then a polynomial in K[X] is naturally an element of L[X] also. A polynomial which is irreducible over K need not continue to remain so over L. For example, the polynomial x² – 2 is irreducible over , but reducible over , since , being a real number but not a rational number. As a second example, the polynomial x² + 1 is irreducible over both and but not over . In fact, we will show shortly that an irreducible polynomial in K[X] of degree > 1 becomes reducible over a suitable extension of K.

For polynomials f(X), with g(X) ≠ 0, there exist unique polynomials q(X) and r(X) in K[X] such that f(X) = q(X)g(X) + r(X) with r(X) = 0 or deg r(X) < deg g(X). The polynomials q(X) and r(X) are respectively called the quotient and remainder of polynomial division of f(X) by g(X) and can be obtained by the so-called long division procedure. We use the notations: q(X) = f(X) quot g(X) and r(X) = f(X) rem g(X).

Whenever we talk about the gcd of two non-zero polynomials, we usually refer to the monic gcd, that is, a polynomial with leading coefficient 1. This makes the gcd of two polynomials unique. We have gcd(f(X), g(X)) = gcd(g(X), r(X)), where r(X) = f(X) rem g(X). This gives rise to an algorithm (similar to the Euclidean gcd algorithm for integers) for computing the gcd of two polynomials. Bézout relations also hold for polynomials. More specifically:

Proposition 2.23.

The concept of congruence can be extended to polynomials, namely, if , then two polynomials g(X), are said to be congruent modulo f(X), denoted g(X) ≡ h(X) (mod f(X)), if f(X)|(g(X) – h(X)), that is, if there exists with g(X) – h(X) = u(X)f(X), or equivalently, if g(X) rem f(X) = h(X) rem f(X).

The principal ideals 〈f(X)〉 of K[X] play an important role (as do the ideals 〈n〉 of ). Let us investigate the structure of the quotient ring R := K[X]/〈f(X)〉 for a non-constant polynomial . If r(X) denotes the remainder of division of by f(X), then it is clear that the residue classes of g(X) and r(X) are the same in R. On the other hand, two polynomials g(X), with deg g(X) < deg f(X) and deg h(X) < deg f(X) represent the same residue class in R if and only if g(X) = h(X). Thus elements of R are uniquely representable as polynomials of degrees < deg f(X). In other words, we may represent the ring R as the set together with addition and multiplication modulo the polynomial f(X). The ring R contains all the constant polynomials , that is, the field K is canonically embedded in R. In general, R is not a field. The next theorem gives the criterion for R to be a field.

Theorem 2.23.

Let L := K[X]/〈f(X)〉 with f(X) irreducible over K. Then K ⊆ L is a field extension. If deg f(X) = 1, then L is isomorphic to K. If deg f(X) ≥ 2, then L is a proper extension of K. This gives us a useful and important way of representing the extension field L, given a representation for K. (For example, see Section 2.9.)

2.6.2. Roots of Polynomials

The study of the roots of a polynomial is the central objective in algebra. We now derive some elementary properties of roots of polynomials.

Definition 2.36.

Proposition 2.24.

Proposition 2.25.

In the last proof, the only result we have used to exploit the fact that K is a field is that K contains no non-zero zero divisors. This is, however, true for every integral domain. Thus Proposition 2.25 continues to hold if K is any integral domain (not necessarily a field). However, if K is not an integral domain, the proposition is not necessarily true. For example, if ab = 0 with a, , a ≠ b, then the polynomial X² + (b – a)X has at least three roots: 0, a and a – b.

For a field extension K ⊆ L and for a polynomial , we may think of the roots of f in L, since too. Clearly, all the roots of f in K are also roots of f in L. However, the converse is not true in general. For example, the only roots of X⁴ – 1 in are ±1, whereas the roots of the same polynomial in are ±1, ±i. Indeed we have the following important result.

Proposition 2.26.

We say that the field K′ in the proof of the last proposition is obtained by adjoining the root α of f and denote this as K′ = K(α). We can write f(X) = (X – α)f₁(X), where and deg f₁ = (deg f) – 1. Now there is a field extension K″ of K′, where f₁ has a root. Proceeding in this way we prove the following result.

Proposition 2.27.

If a polynomial of degree d ≥ 1 has all its roots α₁, . . . , α_d in L, then f(X) = a(X – α₁) · · · (X – α_d) for some (actually ). In this case, we say that f splits (completely or into linear factors) over L.

Definition 2.37.

Every non-constant polynomial has a splitting field L over K. Quite importantly, this field L is unique in some sense. This allows us to call the splitting field of f instead of a splitting field of f. We discuss these topics further in Section 2.8.

Definition 2.38.

The notion of multiplicity can be extended to a non-root β of f by setting the multiplicity of β to zero.

2.6.3. Algebraic Elements and Extensions

Here we assume, unless otherwise stated, that K ⊆ L is a field extension.

Definition 2.39.

Example 2.10.

Definition 2.40.

Proposition 2.28.

By Proposition 2.28, a monic minimal polynomial f of α over K is uniquely determined by α and K. It is, therefore, customary to define the minimal polynomial of α over K to be this (unique) monic polynomial. Unless otherwise stated, we will stick to this revised definition and write f(X) = minpoly_{α, K}(X).

Example 2.11.

Proposition 2.29.

Definition 2.41.

We will see in Section 2.8 that an algebraic closure of every field exists and is unique in some sense. The algebraic closure of an algebraically closed field K is K itself. We end this section with the following well-known theorem. We will not prove the theorem in this book, because every known proof of it uses some kind of complex analysis which this book does not deal with.

Theorem 2.24. Fundamental theorem of algebra

Exercise Set 2.6

2.51	Let R be a ring and f, . Show that: deg(f + g) ≤ max(deg f, deg g) with equality holding, if deg f ≠ deg g. deg(f g) ≤ deg f + deg g with equality holding, if R is an integral domain. If R is an integral domain, then R[X] is an integral domain too. More generally, if R is an integral domain, then R[X1, . . . , Xn] is also an integral domain for all .
2.52	Let f, , where R is an integral domain. Show that if f(ai) = g(ai) for i = 1, . . . , n, where n > max(deg f, deg g) and where a1, . . . , an are distinct elements of R, then f = g. In particular, if f(a) = g(a) for an infinite number of , then f = g.
2.53	Lagrange’s interpolation formula Let K be a field and let a0, . . . , an be distinct elements of K. Show that for (not necessarily all distinct), there exists a unique polynomial of degree ≤ n such that f(ai) = bi for all i = 0, . . . , n. [H]
2.54	Polynomials over a UFD Let R be a UFD. For a non-zero polynomial , a gcd of the coefficients of f is called a content of f and is denoted by cont f. One can then write f = (cont f)f1, where with cont . f1 is called a primitive part of f and is often denoted as pp f. It is clear that cont f and pp f are unique up to multiplication by units of R. If for a non-zero polynomial the content cont (or, equivalently, if f and pp f are associates), then f is called a primitive polynomial. Show that for two non-zero polynomials f, the elements cont(f g) and (cont f)(cont g) are associates in R. In particular, the product of two primitive polynomials is again primitive.
2.55	Let R be a UFD. Show that a non-constant polynomial is irreducible over R if and only if f is irreducible over Q(R), where Q(R) denotes the quotient field of R (see Exercise 2.34).
2.56	Eisenstein’s criterion Let R be a UFD and with an ≠ 0. Suppose that there is a prime such that p does not divide an, p divides ai for all i, 0 ≤ i ≤ n – 1, and p2 does not divide a0. Show that f is irreducible over R. As an application of Eisenstein’s criterion show that for a prime the polynomial Xp–1 + · · · + X + 1 is irreducible in . [H]
2.57	Let K ⊆ L be a field extension and f1, . . . , fn non-constant polynomials in K[X]. Show that each fi, i = 1, . . . , n, splits over L if and only if the product f1 · · · fn splits over L.
2.58	Show that the irreducible polynomials in have degrees ≤ 2. [H]
2.59	Show that a finite field (that is, a field with finite cardinality) is not algebraically closed. In particular, the algebraic closure of a finite field is infinite.
2.60	A complex number z is called an algebraic number, if z is algebraic over . An algebraic number z is called an algebraic integer, if z is a root of a monic polynomial in . Show that: If z is an algebraic number, then mz is an algebraic integer for some . If is an algebraic integer, then . If is an algebraic integer, then for any integer the complex numbers nz and z + n are algebraic integers.
2.61	Let K be a field and . The formal derivative f′ of f is defined to be the polynomial . Show that: (f + g)′ = f′ + g′ and (f g)′ = f′g + f g′ for any f, . If char K = 0, then f′ = 0 if and only if . If char K = p > 0, then f′ = 0 if and only if f(X) = g(Xp) for some . f (≠ 0) has no multiple roots (in any extension field of K), that is, f is square-free, if and only if gcd(f, f′) = 1. Let f be a (non-constant) irreducible polynomial over K. Show that if char K = 0, then f has no multiple roots. On the other hand, if char K = p > 0, show that f has multiple roots if and only if f(X) = g(Xp) for some . (However, if , then by Fermat’s little theorem g(Xp) = g(X)p, which contradicts the fact that f(x) is irreducible. Therefore, f cannot have multiple roots.)
2.62	Let be a non-constant polynomial of degree d and let α1, . . . , αd be the roots of f (in some extension field of K). The quantity is called the discriminant of f. Prove the following assertions: Δ(f) = 0 if and only if f has a multiple root. . Δ(X2 + aX + b) = a2 – 4b. Δ(X3 + aX + b) = –(4a3 + 27b2).