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An Introduction to the Theory of Field Extensions

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An Introduction to the Theory of Field Extensions AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS SAMUEL MOY Abstract. Assuming some basic knowledge of groups, rings, and fields, the following investigation will introduce the reader to the theory of rings before proceeding to elaborate, in greater depth, on the theory of field extensions. Finally, a few consequences of the subject will be examined by solving classical straightedge and compass problems in a manner that effectively utilizes the material. Contents 1. The Basics 1 2. Ring Theory 1 3. Fields and Field Extensions 4 4. Algebraic Field Extensions 9 5. Classical Straightedege and Compass Constructions 12 6. Acknowledgements 15 7. References 15 1. The Basics Definition 1.1. :A ring R is a set together with two binary operations + and × (addition and multiplication, respectively) satisyfing the following axioms: (i) (R, +) is an abelian group, (ii) × is associative: (a × b) × c = a × (b × c) for all a; b; c 2 R, (iii) the distributive laws hold in R for all a; b; c 2 R: (a + b) × c = (a × c) + (b × c) and a × (b + c) = (a × b) + (a × c): Definition 1.2. The ring R is commutative if multiplication is commutative. Definition 1.3. The ring R is said to have an identity (or contain a 1) if there is an element 1 2 R with 1 × a = a × 1 = a for all a 2 R Definition 1.4. A ring R with identity 1, where 1 6= 0, is called a division ring (or skew field) if 8 nonzero element a 2 R, 9 b 2 R such that ab = ba = 1. Definition 1.5. A commutative division ring is called a field. Example 1.6. Z is a commutative ring with 1(identity). Q, R, C, and Z/pZ (the integers modulo p, where p is prime) are all fields. 1 2 SAMUEL MOY 2. Ring Theory Before beginning further study of fields, additional knowledge pertaining to rings is necessary. Most notably, the topics of ideals, ring homomorphisms and isomor- phisms, and quotient rings must first be approached. Definition 2.1. A subring of the ring R is a subgroup of R that is closed under multiplication (i.e. A subset S of a ring R is a subring if the operations of addition and multiplication in R when restricted to S give S the structure of a ring). Definition 2.2. Let R and S be rings. (1) A ring homomorphism is a map ' : R ! S satisfying (a) '(a + b) = '(a) + '(b) for all a; b 2 R, and (b) '(ab) = '(a)'(b) for all a; b 2 R. (2) The kernel of the ring homomorphism ', denoted ker', is the set of ele- ments of R that map to 0 in S. (3) A bijective ring homomorphism is called an isomorphism. Definition 2.3. Let R be a ring, let I be a subset of R and let r 2 R. (1) rI = fra j a 2 Ig and Ir = far j a 2 Ig. (2) A subset I of R is a left ideal (respectively right ideal) of R if (a) I is a subring of R, and (b) I is closed under left multiplication (respectively right multiplication) by elements from R. (3) A subset I that is both a left ideal and a right ideal is called an ideal (or, a two-sided ideal) of R. Definition 2.4. Let R be a ring and I an ideal of R. Then the quotient ring of R by I, denoted R=I is the ring defined by the following binary operations: (r + I) + (s + I) = (r + s) + I and (r + I) × (s + I) = (rs + I) 8r; s 2 R: Theorem 2.5. (The First Isomorphism Theorem for Rings) (1) If ' : R ! S is a homomorphism of rings, then the kernel of ' is an ideal of R, the image of ' is a subring of S and R/ker' is isomorphic as a ring to '(R). (2) If I is any ideal of R, then the map R ! R=I defined by r 7! r + I is a surjective ring homomorphism with kernel I (this homomorphism is called the natural projection of R onto R/I). Thus, every ideal is the kernel of a ring homomorphism and vice versa. Proof. Let I be the kernel of '. Then the cosets under addition of I are exactly the fibers of ' (the sets of elements of R that map to a single element of S). In particular, the cosets r+I; s+I, and rs+I are the fibers of ' over '(r);'(s);'(rs), respectively. Since ' is a ring homomorphism, '(r)'(s) = '(rs), hence (r + I)(s + I) = rs + I. Multiplication of cosets is well defined and so I is an ideal and R=I is a ring. The correspondance r + I 7! '(r) is a bijection between the rings R=I and '(R) which respects addition and multiplication. Hence, it is a ring isomorphism. If I is any ideal, then R=I is a ring (in particular is an abelian group) and the AN INTRODUCTION TO THE THEORY OF FIELD EXTENSIONS 3 map π : r 7! r + I is a group homomorphism with kernel I (natural projection for groups). It remains to check that π is a ring homomorphism. This is immediate from the definition of multiplication in R=I: π : rs 7! rs + I = (r + I)(s + I) = π(r)π(s) Theorem 2.6 (The Lattice Isomorphism Theorem for Rings). Let R be a ring and let I be an ideal of R. The correspondance A $ A=I is an inclusion preserving bijection between the set of subrings A of R that contain I and the set of subrings of R=I. Furthermore, A (a subring containing I) is an ideal of R if and only if A=I is an ideal of R=I. The proof for this theorem will not be provided. However, it follows almost immediately from the Lattice Isomorphism Theorem for Groups. (In addition, rather than considering subgroups, we must consider ideals). Definition 2.7. Let I and J be ideals of R. (1) Define the sum of I and J by I + J = fa + b j a 2 I; b 2 Jg. (2) Define the product of I and J, denoted by IJ, to be the set of all finite sums of elements of the form ab with a 2 I and b 2 J. Definition 2.8. Let A be any subset of the ring R. Let (A) denote the smallest ideal of R containing A, called the ideal generated by A, \ (A) = I: I⊇A Definition 2.9. An ideal M in an arbitrary ring R is called a maximal ideal if M 6= R and the only ideals containing M are M and R. Definition 2.10. Assume R is a commutative ring. An ideal P is called a prime ideal if P 6= R and whenever the product ab of two elements a; b 2 R is an element of P , then at least one of a and b is an element of P . The following two propositions will be useful for later theorems regarding fields. Only Proposition 2.12 will be proved now (Proposition 2.11 will appear as a Lemma for a later theorem and will be proved then). Proposition 2.11. Assume R is a commutative ring. Then R is a field if and only if its only ideals are 0 and R. Proposition 2.12. Assume R is a commutative ring. The ideal M is a maximal ideal if and only if the quotient ring R=M is a field. Proof. This follows from the Lattice Isomorphism Theorem for Rings along with Proposition 2.11. The ideal M is maximal if and only if there are no ideals I with M ⊂ I ⊂ R. By the Lattice Isomorphism Theorem the ideals of R containing M correspond bijectively with the ideals of R=M, so M is maximal if and only if the ideals of R=M are 0 and R=M. By Proposition 2.11 we see that M is a maximal ideal if and only if R=M is a field. 4 SAMUEL MOY Example 2.13. To assist in the understanding of what an ideal is, we will consider ideals in the commutative ring with 1, Z. An example of an ideal in this ring is (2) = f2a j a 2 Zg = multiples of 2 (the ideal generated by the element 2 2 Z). Now, it should not be difficult to see that the only prime ideals in Z are those generated by prime numbers: Suppose we have the ideal (n), n composite (and positive) (i.e. n = ab, 0 < a; b 6= 1). WLOG, assume a < b < n. Clearly a · n 2 (n) ) a · a · b = a2 · b 2 (n). However, 0 < a2 < n and 0 < b < n ) neither a2 nor b 2 (n). Therefore, (n) is not a prime ideal. Now take the ideal (p), p prime. Suppose ab 2 (p). Then p j ab ) p j a or p j b (in other words, a = n · p, n 2 N or b = n · p, n 2 N) ) a 2 (p) or b 2 (p) ) (p) must be a prime ideal. In addition, we see that the only maximal ideals in Z generated by a single element are, again, those generated by prime numbers. To see why this is true, choose some ideal (p), p prime. The only ideals that contain the element p 2 Z are (1) and (p). But (1) = Z! Thus, (p) must then be a maximal ideal.
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