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Abstract Algebra and Algebraic Number Theory

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Abstract Algebra and Algebraic Number Theory Abstract Algebra and Algebraic Number Theory Shashank Singh January 23, 2011 Contents 1 Introduction 2 2 Basic Algebraic Structures 3 2.1 Group . .3 2.2 Ring and Integral Domain . .7 2.3 Arithmetic in Rings . 14 2.4 DomainsfED,PID,UFDg ..................... 15 3 Field Extensions 17 3.1 Algebraic Extension . 18 3.2 Splitting Field and Algebraic Closure . 19 3.3 Separable Extensions . 21 3.4 Normal Extensions . 22 3.5 Galois Extension . 23 4 Algebraic Number Theory 27 4.1 Algebraic Number and Algebraic Integer . 27 4.2 Norms, Traces and Discriminants . 28 4.2.1 Discriminant . 30 4.3 Dedekind Domain . 31 4.3.1 Unique Factorization of Ideals . 32 4.4 Factorization of Primes in Extensions . 32 4.5 Norm of an Ideal . 33 4.6 Ideal Class Group . 34 1 Chapter 1 Introduction The word algebra stems out from the name of a famous book "Al-Jabr wa-al- Muqabalah" by an Arab Mathematician Alkarismi. Alkarismi lived around the year 800 A.D. In this book he described the basic algebraic techniques to simplify algebraic equations. In the Modern abstract algebra, we study the algebraic structures such as groups, rings and fields in the axiomatic and structured way. Modern abstract algebra arises in attempts to solve the polynomial equa- tions. There were exact methods to solve the polynomial equations of degree up to 4. These methods reduce the polynomial into a lower degree auxiliary equation(s), known as resolvent equation(s). Resolvent equations are then solved using existing methods. Lagrange tried to solve quintic during 1770. While analyzing the quintic Lagrange found that the resolvent equation is of degree six. He did not succeed. Later, Ruffini and Abel proved the unsolvability of the quintic using the ideas of Lagrange resolvent. It was Galois, however, who made the fundamental conceptual advances, and who is considered by many as the founder of group theory. Galois described group as a collection of permutations closed under multiplication. In this short note, we will discuss basic concepts of the Group Theory and Field Theory and using that we will try to cover some aspects of algebraic number theory. Though the Galois's group concept was slightly different than that we see now. We will discuss it in more structured and simpler way. For knowing more about the history of Abstract Algebra, please go through the book [4]. See the books [3], [1] and [2] for more details of abstract algebra and algebraic number theory. 2 Chapter 2 Basic Algebraic Structures 2.1 Group Definition 2.1.1 (Binary Operation). Let G be a set. A binary operation on G is a function o : GXG 7! G. Definition 2.1.2 (Group). Let G be a non empty set together with a binary operation o. We say that (G; o) is a group if the following properties are satisfied. • Associativity. The binary operation o is associative. i:e: (aob) oc = ao (boc) 8a; b; c 2 G. • Identity. There is an element e 2 G, called identity element of the group G, s:t: aoe = a8a 2 G. • Inverse. For each element a in G, there is an element b 2 G, called inverse of a in G, s:t: aob = e = boa. Note. In addition, if the above binary operation is commutative, i:e aob = boa, then we call that group Abelian group or Commutative Group. Example 2.1.1. (Z; +), (Q; +), (R; +), (C; +), (Q; ∗), (R; ∗), (C; ∗) are infinite abelian groups. (Zm; +) is a finite abelian group. (Zp; ∗) is a finite abelian group, where p is a prime integer. Hamiltonian Group Q8 = f1; −1; i; −i; j; −j; k; −kg is a non-abelian group. Definition 2.1.3 (Subgroup). If (G; o) is a group and H is a nonempty subset of G. We say that H is a subgroup of G, if (H; o) is itself a Group. 3 Proposition 2.1.1. Consider a group (Z; +). A non-empty subset H of Z is a subgroup of (Z; +) iff H = mZ for some m 2 Z. Subgroup Generated by a subset. Let S be a subset of a group (G; o).Then intersection of all those subgroups of G, which contains S is also a subgroup of G. This smallest subgroup of G containing the subset S is called a subgroup generated by S and is denoted by [S]. Example 2.1.2. [φ] = feg,[G] = G. Definition 2.1.4 (Cyclic Group). A group (G; o) is called a cyclic group r if 9a 2 G s:t: G = [fag] = fa : r 2 Zg. Example 2.1.3. (Z; +) is an infinite cyclic group generated by 1 of −1. (Zn; +) is finite cyclic group generated by 1. Note. Generator of a cyclic group may not be unique. The order of a group G is simply the number of elements in G.The order of an element g in a group is the least positive integer k such, that gk is the identity if there is such a number k, or infinite otherwise. Definition 2.1.5 (Coset Decomposition). Let H be a subgroup of G. Let a 2 G, then aH = faoh : h 2 Hg is called left coset of H in G determined by a. Ha = fhoa : h 2 Hg is called right coset of H in G determined by a. Properties • (aH = H) , a 2 H. • (aH = bH) , a−1b 2 H • (aH = bH) , b−1a 2 H • If G is a finite group, then, o (H) = o (aH). • Number of left cosets of H equals number of right coset H. Definition 2.1.6 (Index of a subgroup). If a group G is finite, then the number of left cosets of a subgroup H of a G is called the index of H in G, denoted by [G : H]. The set of left cosets of H in G is denoted by G . =H G = faH : a 2 Gg =H It is called quotient set of G by the subgroup H. 4 Theorem 2.1.2 (Lagrange). The order and index of a subgroup of a finite group divide the order of a group.In other words, if H is a subgroup of a finite group G, then, o (G) = o (H) : [G : H] Corollary. Every group of a prime order is cycle and hence abelian. Definition 2.1.7 (Normal Subgroup). A subgroup, N, of a group, G, is called a normal subgroup (denoted by N/G) if family of left cosets is same as family of right cosets; that is, N/G , fgN : g 2 Gg = fNg : g 2 Gg. Remark. If G is a abelian group, then every subgroup of G is normal in G. Theorem 2.1.3. H/G , ghg−1 2 H8a 2 G; 8h 2 H Definition 2.1.8 (Quotient Group). Let H/G, then G = fgH : g 2 Gg =H forms a group with respect to the operation ∗, defined as. xH ∗ yH = (xoy) H for all x; y 2 G This group is called the quotient group of G by the normal subgroup H. Example 2.1.4. Consider (Z; +) and a subgroup mZ. = = fp + m : p 2 g Z mZ Z Z p + mZ = fp + mx : x 2 Zg =p ¯ 2 Zm = = fp¯ : p 2 g Z mZ Z Thus = = m. Z mZ Z Definition 2.1.9 (Simple Group). A group which has no proper normal subgroup is called a simple group. e.g. (Zp; +) is a simple group. Definition 2.1.10 (Maximal Normal Subgroup). Proper normal sub- group H of G is called a maximal normal subgroup of G if K be the normal subgroup of G such that H ⊂ K then either H = K or K = G. i.e. there is no proper normal subgroup between H and G. 5 Definition 2.1.11 (Group Homomorphism). Let (G1;:) and (G2; ∗) be groups, then a map f : G1 7! G2 s.t. f (x:y) = f (x)∗f (y), is called a group homomorphism. • If f is an injective group homomorphism, then it is called a monomor- phism. • If f is an surjective group homomorphism, then it is called a epimor- phism. • If f is an bijective group homomorphism, then it is called a isomor- phism and we write G1 ≈ G2. • A group homomorphism f : G1 7! G1 s.t. f (x:y) = f (x) :f (y) is called an indomorphism and if f is an isomorphism, then it is called as automorphism. We define the kernel of f to be the set of elements in G1 which are mapped to the identity in G2. ker(f) = fg 2 G1 : f(g) = e2g and the image of f to be im(f) = ff(g): g 2 G1g The kernel is a normal subgroup of G1 and and the image is a subgroup of G2. Theorem 2.1.4. Every infinite cyclic group is isomorphic to (Z; +). Theorem 2.1.5. Every finite cyclic group of order m is isomorphic to (Zm; +). Proposition 2.1.6. Let (G; o) be a group, then Aut (G) = ff : fis an automorphism on G g forms a group with respect to composition of maps. Theorem 2.1.7 (Fundamental Theorem of group homomorphism). Let G1 and G2 be two groups and f : G1 7! G2 be a surjective group homo- morphism, then G ≈ G 1=ker(f) 2 6 Note (Survey of Groups upto order 7). We know that every finite group of prime order is cyclic and two finite cyclic group of same order are isomor- phic.
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