Notes for 4H Galois Theory 2002–3 Andrew Baker [05/02/2003] Department of Mathematics, University of Glasgow. E-mail address: [email protected] URL: http://www.maths.gla.ac.uk/∼ajb √ Q( 3 2,ζ ) Q(√(cid:78)(cid:78)3 2) (cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)(cid:80)Q(cid:103)(cid:80)(cid:103)(cid:80)(cid:103)((cid:80)(cid:103)(cid:80)√(cid:103)3(cid:80)(cid:103)(cid:80)(cid:78)(cid:78)2(cid:103)(cid:80)(cid:103)(cid:80)23ζ(cid:103)(cid:80)(cid:68)(cid:103)(cid:80)3(cid:68)(cid:103)(cid:80))(cid:68)(cid:80)(cid:103)(cid:68)(cid:109)(cid:80)(cid:103)(cid:68)(cid:109)(cid:68)(cid:80)(cid:103)(cid:68)(cid:109)(cid:80)(cid:103)(cid:68)(cid:109)(cid:80)(cid:103)(cid:68)(cid:109)(cid:80)(cid:68)(cid:103)(cid:109)(cid:80)(cid:68)(cid:103)(cid:109)2(cid:68)(cid:80)(cid:103)3(cid:68)(cid:109)(cid:80)(cid:103)(cid:68)(cid:109)(cid:80)(cid:68)(cid:103)(cid:109)(cid:80)(cid:68)(cid:103)(cid:109)(cid:80)(cid:68)(cid:109)(cid:68)(cid:80)(cid:68)Q(cid:80)(cid:68)(cid:80)(cid:68)(cid:80)(cid:68)((cid:80)(cid:68)√3(cid:73)(cid:73)(cid:74)(cid:74) 223(cid:106)ζ(cid:106)32(cid:73)3(cid:106)(cid:73))(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)2(cid:106)(cid:73)(cid:73)(cid:106)3(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:106)(cid:73)(cid:73)Q(ζ(cid:81)(cid:81) 3) Q (cid:83)(cid:83) (cid:175)(cid:175) ∼ Gal(E/Q) = S 3 {id,((cid:179)(cid:179)2 3)}(cid:110)(cid:110)(cid:110)(cid:88)(cid:110)(cid:88)(cid:110)(cid:88)(cid:110)(cid:88)(cid:110)(cid:88)(cid:110)(cid:88)(cid:110){(cid:88)(cid:110)(cid:88)(cid:110)i(cid:88)(cid:110)d(cid:88)(cid:110),(cid:110)(cid:88)(cid:110)(cid:88)((cid:110)(cid:88)(cid:179)(cid:179)1(cid:110)3(cid:88)2(cid:110)(cid:88)(cid:110)3(cid:88)(cid:122)(cid:110)(cid:88)(cid:122))(cid:110)(cid:122)(cid:88)}(cid:110)(cid:122)(cid:88)(cid:81)(cid:110)(cid:122)(cid:81)(cid:88)(cid:122)(cid:110)(cid:81)(cid:88)(cid:122)(cid:110)(cid:122)(cid:81)(cid:88)(cid:110)(cid:122)(cid:81)(cid:88)(cid:110)3(cid:122)(cid:81)2(cid:110)(cid:88)(cid:122)(cid:81)(cid:110)(cid:88)(cid:122)(cid:81)(cid:110)(cid:122)(cid:88)(cid:81)(cid:122)(cid:110)(cid:88)(cid:81)(cid:122)(cid:110)(cid:88)(cid:122)(cid:81)(cid:110)(cid:88)(cid:122)(cid:81)(cid:110)(cid:122)(cid:88){(cid:81)(cid:110)(cid:122)(cid:88)(cid:81)(cid:122)i(cid:88)(cid:81)(cid:122)d(cid:122),32((cid:184)(cid:184)(cid:183)(cid:183) 1 2(cid:117))(cid:117)}(cid:117)(cid:117)(cid:117)(cid:84)(cid:117)(cid:84)(cid:117)(cid:84)(cid:117)(cid:84)(cid:117)(cid:84)(cid:117)(cid:84)(cid:117)(cid:84)(cid:117)3(cid:84)(cid:117)2(cid:84)(cid:117)(cid:117)(cid:84)(cid:117)(cid:84){(cid:117)(cid:84)(cid:117)i(cid:84)(cid:117)d(cid:84)(cid:117)(cid:84)(cid:117),(cid:84)(cid:117)((cid:117)(cid:117)1(cid:117)(cid:117)2 3)(cid:176)(cid:176) ,(1 3 2)} {id} √ The Galois Correspondence for Q( 3 2,ζ )/Q 3 ii Introduction: What is Galois Theory? Much of early algebra centred around the search for explicit formulæ for roots of polynomial equations in one or more unknowns. The solution of linear and quadratic equations in one unknownwaswellunderstoodinantiquity,whileformulæfortherootsofgeneralrealcubicsand quartics was solved by the 16th century. These solutions involved complex numbers rather than justrealnumbers. Bytheearly19thcenturynogeneralsolutionofageneralpolynomialequation ‘by radicals’ (i.e., by repeatedly taking n-th roots for various n) was found despite considerable effort by many outstanding mathematicians. Eventually, the work of Abel and Galois led to a satisfactory framework for fully understanding this problem and the realization that the general polynomial equation of degree at least 5 could not always be solved by radicals. At a more profound level, the algebraic structure ofGalois extensions is mirrored in the subgroups of their Galois groups, which allows the application of group theoretic ideas to the study of fields. This Galois Correspondence is a powerful idea which can be generalized to apply to such diverse topics as ring theory, algebraic number theory, algebraic geometry, differential equations and algebraic topology. Because of this, Galois theory in its many manifestations is a central topic in modern mathematics. In this course we will focus on the following topics. • Thesolutionofpolynomialequationsoverafield,includingrelationshipsbetweenroots, methods of solutions and location of roots. • The structure of finite and algebraic extensions of fields and their automorphisms. We will study these in detail, building up a theory of algebraic extensions of fields and their automorphism groups and applying it to questions about roots of polynomial equations. The techniques we discuss can also be applied to the following topics some of which will be met by students taking advanced degrees. • Classical topics such as ‘squaring the circle’, ‘duplication of the cube’, constructible numbers and constructible polygons. • Applications of Galois theoretic ideas in Number Theory, the study of differential equations and Algebraic Geometry. There are many good introductory books on Galois Theory and we list some of them in the bibliography at the end. Suggestions on using these notes These notes cover more than the content of the course and should be used in parallel with the lectures. The problem sets contain samples of the kind of problems likely to occur in the final examination and should be attempted as an important part of the learning process. ♠ The symbol ♥ ♦ means the adjacent portion of the notes is not examinable. ♣ Contents Introduction: What is Galois Theory? ii Suggestions on using these notes ii Chapter 1. Integral domains, fields and polynomial rings 1 1.1. Recollections on integral domains and fields 1 1.2. Polynomial rings 5 1.3. Identifying irreducible polynomials 10 1.4. Finding roots of complex polynomials of small degree 13 1.5. Automorphisms of rings and fields 16 Exercises on Chapter 1 20 Chapter 2. Fields and their extensions 23 2.1. Fields and subfields 23 2.2. Simple and finitely generated extensions 25 Exercises on Chapter 2 28 Chapter 3. Algebraic extensions of fields 29 3.1. Algebraic extensions 29 3.2. Splitting fields and Kronecker’s Theorem 32 3.3. Monomorphisms between extensions 35 3.4. Algebraic closures 37 3.5. Multiplicity of roots and separability 40 3.6. The Primitive Element Theorem 43 3.7. Normal extensions and splitting fields 45 Exercises on Chapter 3 45 Chapter 4. Galois extensions and the Galois Correspondence 47 4.1. Galois extensions 47 4.2. Working with Galois groups 47 4.3. Subgroups of Galois groups and their fixed fields 49 4.4. Subfields of Galois extensions and relative Galois groups 50 4.5. The Galois Correspondence 51 4.6. Galois extensions inside the complex numbers and complex conjugation 53 4.7. Kaplansky’s Theorem 54 Exercises on Chapter 4 56 Chapter 5. Galois extensions for fields of positive characteristic 59 5.1. Finite fields 59 5.2. Galois groups of finite fields and Frobenius mappings 62 5.3. The trace and norm mappings 64 Exercises on Chapter 5 65 Chapter 6. A Galois Miscellany 67 6.1. A proof of the Fundamental Theorem of Algebra 67 6.2. Cyclotomic extensions 67 6.3. Artin’s Theorem on linear independence of characters 71 6.4. Simple radical extensions 73 iii iv CONTENTS 6.5. Solvability and radical extensions 74 6.6. Galois groups of even and odd permutations 77 6.7. Symmetric functions 79 Exercises on Chapter 6 80 Bibliography 83 Solutions 85 Chapter 1 85 Chapter 2 91 Chapter 3 93 Chapter 4 95 Chapter 5 97 Chapter 6 98 CHAPTER 1 Integral domains, fields and polynomial rings In these notes, a ring will always be a ring with unity 1 which is assumed to be non-zero, 1 (cid:54)= 0. In practice, most of the rings encountered will also be commutative. An ideal I (cid:47)R always means a 2-sided ideal. An ideal I (cid:47)R in a ring R is proper if I (cid:54)= R, or equivalently if I (cid:32) R. For a ring homomorphism ϕ: R −→ S, the unity of R is sent to that of S, i.e., ϕ(1) = 1. Definition 1.1. Let ϕ: R −→ S be a ring homomorphism. • ϕ is a monomorphism if it is injective. • ϕ is an epimorphism if it is surjective. • ϕ is an isomorphism if it is both a monomorphism and an epimorphism. Remark 1.2. The following are equivalent formulations of the notions in Definition 1.1. • ϕ is a monomorphism if and only if for r ,r ∈ R, if ϕ(r ) = ϕ(r ) then r = r , or 1 2 1 2 1 2 equivalently, if r ∈ R with ϕ(r) = 0 then r = 0. • ϕ is an epimorphism if and only if for every s ∈ S there is an r ∈ R with ϕ(r) = s. • ϕ is an isomorphism if and only if it is invertible (and whose inverse is also an isomor- phism). 1.1. Recollections on integral domains and fields The material in this section is standard and most of it should be familiar. For details see [3, 4] or any other book containing introductory ring theory. Definition 1.3. A commutative ring R in which there are no zero-divisors is called an integral domain or an entire ring. This means that for u,v ∈ R, uv = 0 =⇒ u = 0 or v = 0. Example 1.4. The following rings are integral domains. (i) The ring of integers, Z. (ii) If p is a prime, the ring of integers modulo p, F = Z/p = Z/(p). p (iii) The rings of rational numbers, Q, real numbers, R, and complex numbers, C. (iv) The polynomial rings Z[X], Q[X], R[X] and C[X]. Definition 1.5. Let I (cid:47)R be a proper ideal in a commutative ring R. • I is a prime ideal if for u,v ∈ R, uv ∈ I =⇒ u ∈ I or v ∈ I. Similarly, I is a maximal ideal R if whenever J (cid:47)R is a proper ideal and I ⊆ J then J = I. • I (cid:47)R is principal if I = (p) = {rp : r ∈ R} for some p ∈ R. Notice that if p,q ∈ R, then (q) = (p) if and only if q = up for some unit u ∈ R. We also write p | x if x ∈ (p). • p ∈ R is prime if (p)(cid:47)R is a prime ideal; this is equivalent to the requirement that whenever p | xy with x,y ∈ R then p | x or p | y. • Risaprincipal ideal domain ifitisanintegraldomainandeveryidealI(cid:47)Risprincipal. The following fundamental example should be familiar. 1 2 1. INTEGRAL DOMAINS, FIELDS AND POLYNOMIAL RINGS Example 1.6. Every ideal I(cid:47)Z is principal, so I = (n) for some n ∈ Z which we can always take to be non-negative, n (cid:62) 0. Hence Z is a principal ideal domain. Proposition 1.7. Let R be a commutative ring and I (cid:47)R an ideal. (i) The quotient ring R/I is an integral domain if and only if I is a prime ideal. (ii) The quotient ring R/I is a field if and only if I is a maximal ideal. Example 1.8. If n (cid:62) 0, the quotient ring Z/n = Z/(n) is an integral domain if and only if n is a prime. For any (not necessarily commutative) ring with unit there is an important ring homomor- phism η: Z −→ R called the unit or characteristic homomorphism which is defined by   1+···+1 if n > 0,  (cid:124) (cid:123)(cid:122) (cid:125)   n η(n) = n1 = −(1+···+1) if n < 0,  (cid:124) (cid:123)(cid:122) (cid:125)    −n  0 if n = 0. Sincetheunit1 ∈ Risnon-zero,kerη(cid:47)Zisanon-zeroidealandusingtheIsomorphismTheorems we see that there is a quotient monomorphism η: Z/kerη −→ R which allows us to identify the quotient ring Z/kerη with the image ηZ ⊆ R as a subring of R. By Example 1.6, there is a unique non-negative integer p (cid:62) 0 such that kerη = (p); this p is called the characteristic of R and denoted charR. Lemma 1.9. If R is an integral domain, its characteristic charR is a prime. Proof. Consider p = charR. If p = 0 we are done. So suppose that p > 0. The quotient monomorphism η: Z/kerη −→ R identifies Z/kerη with the subring imη = imη of the integral domainR. Buteverysubringofanintegraldomainisitselfanintegraldomain, henceZ/kerη is anintegraldomain. NowbyProposition1.7(i), kerη = (p)isprimeidealandsobyExample1.8, p is a prime. (cid:164) Remark 1.10. When discussing a ring with unit R, we can consider it as containing as a subring of the form Z/(charR) since the quotient homomorphism η: Z/(charR) −→ R gives an isomorphism Z/(charR) −→ imη, allowing us to identify these rings. In particular, every integraldomaincontainsasasubringeitherZ = Z/(0)(ifcharR = 0)orZ/(p)ifp = charR > 0 is a non-zero prime. This subring is sometimes called the characteristic subring of R. The rings Z and Z/n = Z/(n) for n > 0 are often called core rings. When considering integral domains, the rings Z and F = Z/p = Z/(p) for p > 0 a prime are called prime rings. p Here is a useful and important fact about rings containing a finite prime ring. Theorem 1.11 (Idiot’s Binomial Theorem). Let R be a commutative ring with unit con- taining F for some prime p > 0. If u,v ∈ R, then p (u+v)p = up+vp. Proof. We have p1 = 0 in R. Now the Binomial Theorem gives (cid:181) (cid:182) (cid:181) (cid:182) (cid:181) (cid:182) p p p (u+v)p = up+ up−1v+ up−2v2+···+ uvp−1+vp. 1 2 p−1 Suppose that 1 (cid:54) j (cid:54) p−1. Then we have (cid:181) (cid:182) p p(p−1)! = . j j!(p−j)! There are no factors of p appearing in (p−1)!, j! or (p−j)!, so since this number is an integer it must be divisible by p, i.e., (cid:181) (cid:182) p (1.1a) p | , j 1.1. RECOLLECTIONS ON INTEGRAL DOMAINS AND FIELDS 3 or equivalently (cid:181) (cid:182) p (1.1b) ≡ 0 (mod p). j Hence in R we have (cid:181) (cid:182) p 1 = 0. j Combining this fact with the above expansion, we obtain the required equation in R, (u+v)p = up+vp. (cid:164) A commutative ring with unit k is a field if every non-zero element u ∈ k is a unit in k i.e., there is an element v ∈ k such that uv = vu = 1. We usually write u−1 for this element v which is necessarily unique and called the multiplicative inverse of u in k. Example 1.12. If n (cid:62) 0, the quotient ring Z/n is a field if and only if n is a positive prime. Proposition 1.13. If k is a field then it is an integral domain. Proof. Suppose that u,v ∈ k and uv = 0. Then if u (cid:54)= 0 we can multiply this equation by u−1 to obtain v = u−1uv = 0, hence v = 0. So at least one of u,v must be 0. (cid:164) A non-zero element p ∈ R is irreducible if whenever p = uv with u,v ∈ R, either u or v is a unit. Lemma 1.14. Let R be an integral domain. If p ∈ R is a non-zero prime then it is an irreducible. Proof. Suppose that p = uv for some u,v ∈ R. Then p | u or p | v, and we might as well assume that u = tp for some t ∈ R. Then (1−tv)p = 0 and so tv = 1, showing that v is a unit with inverse t. (cid:164) Now let D be an integral domain. A natural question to ask is whether D can be found as a subring of a field. This is certainly true for the integers Z which are contained in the field of rational numbers Q, and for a prime p > 0, the prime ring F is itself a field. p Definition 1.15. The fields Q and F where p > 0 is prime are the prime fields. p Of course, we can view Z as a subring of any subfield of the complex numbers so an answer to this question may not be unique! However, there is always a ‘smallest’ such field which is unique up to an isomorphism. Theorem 1.16. Let D be an integral domain. (i) There is a field of fractions of D, Fr(D), which contains D as a subring. (ii) If ϕ: D −→ F is a ring monomorphism into a field F, there is a unique homomorphism ϕ(cid:101): Fr(D) −→ F such that ϕ(cid:101)(t) = ϕ(t) for all t ∈ D ⊆ Fr(D). ϕ D (cid:47)(cid:47) F (cid:60)(cid:60) inc (cid:178)(cid:178) ∃! ϕ(cid:101) Fr(D) ♠ ♥ ♦ Proof. (i) Consider the set ♣ P(D) = {(a,b) : a,b ∈ D, b (cid:54)= 0}. Now introduce an equivalence relation ∼ on P(D), namely (a(cid:48),b(cid:48)) ∼ (a,b) ⇐⇒ ab(cid:48) = a(cid:48)b. 4 1. INTEGRAL DOMAINS, FIELDS AND POLYNOMIAL RINGS Of course, it is necessary to check that this relation is an equivalence relation; this is left as an exercise. We denote the equivalence class of (a,b) by [a,b] and the set of equivalence classes by Fr(D). We will define addition and multiplication on Fr(D) by [a,b]+[c,d] = [ad+bc,bd], [a,b][c,d] = [ac,bd]. We need to verify that these operations are well defined. For example, if [a(cid:48),b(cid:48)] = [a,b] and [c(cid:48),d(cid:48)] = [c,d], then (a(cid:48)d(cid:48)+b(cid:48)c(cid:48))bd = a(cid:48)d(cid:48)bd+b(cid:48)c(cid:48)bd = ab(cid:48)d(cid:48)d+b(cid:48)bcd(cid:48) = (ad+bc)b(cid:48)d(cid:48), and so (a(cid:48)d(cid:48) + b(cid:48)c(cid:48),b(cid:48)d(cid:48)) ∼ (ad + bc,bd), hence addition is well defined. A similar calculation shows that (a(cid:48)c(cid:48),b(cid:48)d(cid:48)) ∼ (ac,bd), so multiplication is also well defined. It is now straightforward to show that Fr(D) is a commutative ring with zero 0 = [0,1] and unit 1 = [1,1]. In fact, as we will soon see, Fr(D) is a field. Let [a,b] ∈ Fr(D). Then [a,b] = [0,1] if and only if (0,1) ∼ (a,b) which is equivalent to requiring that a = 0; notice that for any b (cid:54)= 0, [0,b] = [0,1]. We also have [a,b] = [1,1] if and only if a = b. Now let [a,b] ∈ Fr(D) be non-zero, i.e., a (cid:54)= 0. Then b (cid:54)= 0 and [a,b],[b,a] ∈ Fr(D) satisfy [a,b][b,a] = [ab,ba] = [1,1] = 1, so [a,b] has [b,a] as an inverse. This shows that Fr(D) is a field. We can view D as a subring of Fr(D) using the map j: D −→ Fr(D); j(t) = [t,1] which is a ring homomorphism; it is easy to check that it is a monomorphism. Therefore we may identify t ∈ D with j(t) = [t,1] ∈ Fr(D) and D with the subring imj ⊆ Fr(D). (ii) Consider the function Φ: P(D) −→ F; Φ(a,b) = ϕ(a)ϕ(b)−1. If (a(cid:48),b(cid:48)) ∼ (a,b) then Φ(a(cid:48),b(cid:48)) = ϕ(a(cid:48))ϕ(b(cid:48))−1 = ϕ(a(cid:48))ϕ(b)ϕ(b)−1ϕ(b(cid:48))−1 = ϕ(a(cid:48)b)ϕ(b)−1ϕ(b(cid:48))−1 = ϕ(ab(cid:48))ϕ(b(cid:48))−1ϕ(b)−1 = ϕ(a)ϕ(b(cid:48))ϕ(b(cid:48))−1ϕ(b)−1 = ϕ(a)ϕ(b)−1 = Φ(a,b), so Φ is constant on each equivalence class of ∼. Hence we can define the function ϕ(cid:101): Fr(D) −→ F; ϕ(cid:101)([a,b]) = Φ(a,b). Itisstraightforwardtoverifythatϕ(cid:101)isaringhomomorphismwhichagreeswithϕonthesubring D ⊆ Fr(D). (cid:164) The next three corollaries are left as an exercise. Corollary 1.19 is sometimes said to imply that the construction of Fr(D) is functorial in the integral domain D. Corollary 1.17. If F is a field then F = Fr(F). Corollary 1.18. If D is a subring of a field F, then Fr(D) ⊆ Fr(F) = F is the smallest subfield of F containing D. Corollary 1.19. If D and D are integral domains and ϕ: D −→ D a ring monomor- 1 2 1 2 phism, there is a unique induced ring homomorphism ϕ : Fr(D ) −→ Fr(D ) for which ϕ (t) = ∗ 1 2 ∗ 1.2. POLYNOMIAL RINGS 5 ϕ(t) if t ∈ D ⊆ Fr(D ). 1 1 ϕ D (cid:47)(cid:47) D 1 2 inc inc (cid:178)(cid:178) (cid:178)(cid:178) Fr(D ) (cid:47)(cid:47) Fr(D ) 1 ϕ 2 ∗ Moreover, this construction has the properties • if ϕ: D −→ D and θ: D −→ D are monomorphisms between integral domains then 1 2 2 3 θ ◦ϕ = (θ◦ϕ) as homomorphisms Fr(D ) −→ Fr(D ); ∗ ∗ ∗ 1 3 • for any integral domain D, the identity homomorphism id: D −→ D induces the iden- tity homomorphism (id) = id: Fr(D) −→ Fr(D). ∗ ϕ θ id D (cid:47)(cid:47) D (cid:47)(cid:47) D D (cid:47)(cid:47) D 1 2 3 inc inc inc inc inc (cid:178)(cid:178) (cid:178)(cid:178) (cid:178)(cid:178) (cid:178)(cid:178) (cid:178)(cid:178) Fr(D ) (cid:47)(cid:47) Fr(D ) (cid:47)(cid:47) Fr(D ) Fr(D) (cid:47)(cid:47) Fr(D) 1 ϕ∗ 2 θ∗ 3 id∗ = id Remark 1.20. When working with a field of fractions it is usual to adopt the notation a = a/b = [a,b] b for the equivalence class of (a,b). The rules for algebraic manipulation of such symbols are the usual ones for working with fractions, i.e., a a a b +a b a a a a a a 1 2 1 2 2 1 1 2 1 2 1 2 + = , × = = . b b b b b b b b b b 1 2 1 2 1 2 1 2 1 2 The field of fractions of an integral domain is sometimes called its field of quotients, however as thewordquotientisalsoassociatedwithquotientringsweprefertoavoidusingthatterminology. 1.2. Polynomial rings Let R be a commutative ring. We will make frequent use of the ring R[X] of polynomials over R in an indeterminate X. This consists of elements of form p(X) = p +p X +···+p Xm 0 1 m where m (cid:62) 0 and p ,p ,...,p ∈ R; such p(X) are called polynomials. Addition and multipli- 0 1 m cation in R[X] are defined by (p +p X +···+p Xm)+(q +q X +···+q Xm) = 0 1 m 0 1 m (p +q )+(p +q )X +···+(p +q )Xm), 0 0 1 1 m m and (p +p X +···+p Xm)(q +q X +···+q Xm) = 0 1 m 0 1 m (p q )+(p q +p q )X +···+(p q +p q +···+p q +p q )X2m. 0 0 0 1 1 0 0 m 1 m−1 m−1 1 m 0 Then R[X] is a commutative ring with the constant polynomials 0 and 1 as its zero and unit. We identify r ∈ R with the obvious constant polynomial; this allows us to view R as a subring of R[X] and the inclusion function inc: R −→ R[X] is a monomorphism. More generally, we inductively can define the ring of polynomials in n indeterminates X ,...,X over R, 1 n R[X ,...,X ] = R[X ,...,X ][X ] 1 n 1 n−1 n for n (cid:62) 1. Again there is an inclusion monomorphism inc: R −→ R[X ,...,X ] which sends 1 n each element of R to the corresponding constant polynomial. These polynomial rings have an important universal property.