Polynomials connect algebra to analysis, number theory and geometry. Given a field , we construct the ring of polynomials , some rings and fields built from it, and we see how to enlarge so that it contains roots of polynomials.
4.3.1 The Polynomial Ring
Definition 4.3.1 (Polynomial ring). Given a field , the polynomial ring in one variable over is Here is a formal variable: it is not an element of or of any other set. Formally, a polynomial is a sequence of elements of with only finitely many nonzero terms. Polynomials are added coefficientwise, and multiplied by expanding and collecting powers of using :
Multiplication is not coefficientwise: . One checks from these formulas that is a commutative ring with identity the constant polynomial . The only axiom that needs a computation is associativity of multiplication: for , and , the coefficient of in both and is . The constants form a subring of that we identify with .
Note. The adjective “formal” matters. When we write , we are not thinking of as a real number; is a symbolic expression. However, we can evaluate a polynomial at any element of , or of any field containing : given and , we set . Comparing with the product formula shows and , so evaluation at is a ring homomorphism .
Definition 4.3.2 (Degree). The degree of a nonzero polynomial is the largest with , and is its leading coefficient. The polynomial is monic if its leading coefficient is . Nonzero constants have degree ; by convention .
Lemma 4.3.3 (Degree of a product). If are nonzero, then and . In particular, has no zero divisors.
Proof. Write and with and . In the coefficient of , every term with or vanishes. If , every pair with has or , so . For , the only surviving term is , which is nonzero because has no zero divisors. ∎
Proposition 4.3.4 (Units of ). The units of are exactly the nonzero constants. In particular, is not a field.
Proof. A nonzero constant has inverse . Conversely, if , then and by the lemma. Both degrees are nonnegative, so and is a nonzero constant. For instance, has no inverse in . ∎
4.3.2 The Field of Rational Functions
To make into a field we allow fractions of polynomials, just as we pass from to .
Definition 4.3.5 (Field of rational functions). The field of rational functions over is Addition and multiplication are defined as for ordinary fractions:
The denominators are nonzero by Lemma 4.3.3. As for , one checks that is an equivalence relation (transitivity uses cancellation in , which holds because has no zero divisors), that the operations do not depend on the chosen representatives, and that the ring axioms hold. The inverse of a nonzero element is , so is a field. Identifying with makes a subring of .
This construction generalizes to polynomials in any number of variables, giving the field of rational functions in variables.
Another way to extend is to allow infinite sums.
Definition 4.3.6 (Formal power series). The ring of formal power series over is Addition is coefficientwise, and multiplication is given by the same formula as for polynomials: for and , we set with .
Note. The product formula makes sense because each coefficient is a finite sum: only the pairs satisfy with . This is why these power series are called “formal”: no question of convergence arises. The ring axioms are checked as for , and contains as a subring (polynomials are the power series with only finitely many nonzero coefficients).
Unlike , the ring has many units.
Lemma 4.3.7 (Units of ). A power series has a multiplicative inverse if and only if .
Proof. () If for some , then comparing constant terms gives , so .
() Assume . We construct with one coefficient at a time. The constant term of must be , so we take . For , the coefficient of in must vanish: The right side involves only , so these equations determine inductively, and the resulting satisfies . ∎
Example 4.3.8. The geometric series: in we have . Indeed, the coefficient of in is for , and the constant term is .
4.3.4 Laurent Series
Every nonzero element of can be written as , where is its first nonzero coefficient. The second factor is a unit by Lemma 4.3.7, so to obtain a field we need only invert , that is, allow finitely many negative powers of .
Definition 4.3.9 (Field of Laurent series). The field of Laurent series over is the set of formal series in which only finitely many negative powers of appear. Addition is coefficientwise and multiplication is again with .
If for and for , the sum defining runs only over , a finite range, and for ; so the product is again a Laurent series. As for , this makes a commutative ring containing as a subring.
Proposition 4.3.10 ( is a field). Every nonzero Laurent series has a multiplicative inverse.
Proof. Let with . Then , where has nonzero constant term. By Lemma 4.3.7, has an inverse in . Then . ∎
4.3.5 Roots of Polynomials and Field Extensions
Given a polynomial with , we can ask whether has a root in , and if not, whether we can extend to a larger field containing one.
Definition 4.3.11 (Root). Let be a field extension of and . An element is a root of if . We write for the extension and sometimes .
Example 4.3.12. The polynomial has no roots in , since is irrational. Consider It contains and the root of , and it is closed under addition and under multiplication, since . It is also closed under inverses: for , Here : if , then would give , and if , then . So is a subfield of , and a field extension of containing a root of .
Example 4.3.13. Similarly, has no real roots. The extension is the field of complex numbers, which contains the roots of .
Lemma 4.3.14 (Factor theorem). Let have degree and let . Then for some of degree . In particular, is a root of if and only if for some .
Proof. For we have . If , then and the coefficient of in is . If this gives . Conversely, if , evaluating at gives . ∎
Corollary 4.3.15 (Number of roots). A nonzero polynomial of degree has at most roots in .
Proof. By induction on . A nonzero constant has no roots. Let . If has no root we are done. Otherwise let be a root and write with . If is another root, then with , so . Thus every root of other than is a root of , and has at most roots by induction. ∎
Definition 4.3.16 (Algebraically closed field). A field is algebraically closed if every non-constant polynomial has a root in .
Proposition 4.3.17 (Factorization into linear factors). A field is algebraically closed if and only if every polynomial of degree factors as (the need not be distinct).
Proof. If every such factors in this way, then is a root of , so is algebraically closed. Conversely, let be algebraically closed; we induct on . If , then . If , let be a root of . By the factor theorem, with , and by induction . ∎
Note. The Fundamental Theorem of Algebra states that is algebraically closed; we do not prove it here. Over an algebraically closed field, every non-constant polynomial already has a root, so adjoining roots of polynomials produces no larger fields. The field is called the algebraic closure of : it is algebraically closed, and every element of is a root of a nonzero real polynomial ( is a root of ). By contrast, is very far from algebraically closed: , , , … have no rational roots, and adjoining their roots gives infinitely many different extensions of .