Abstract Algebra and Algebraic Number Theory

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$LATEX and Beamer (Arijit Bishnu). - Indian Statistical Institute$

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 Domains{ED,PID,UFD} . . . . . . . . . . . . . . . . . . . . . 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 equations. 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
Page 1: Abstract Algebra and Algebraic Numb
Page 5 and 6: Proposition 2.1.1. Consider a group
Page 7 and 8: Definition 2.1.11 (Group Homomorphi
Page 9 and 10: a zerodivisor is called a nonzerodi
Page 11 and 12: Proposition 2.2.6. Let f : R 1 ↦
Page 13 and 14: Theorem 2.2.13 (First Isomorphism T
Page 15 and 16: Note. Let R be a ring. • If R is
Page 17 and 18: Remark. Arithmetic properties of an
Page 19 and 20: Theorem 3.0.10. Let φ : F 1 ↦→
Page 21 and 22: Definition 3.2.1. The filed Q(ζ n
Page 23 and 24: Note. Every irreducible polynomial
Page 25 and 26: Theorem 3.5.2. Let E be a splitting
Page 27 and 28: 3. K/E is always Galois with Galois
Page 29 and 30: 3. The element β belongs to a subr
Page 31 and 32: 4.2.1 Discriminant Definition 4.2.2
Page 33 and 34: 3. If I and J are fractional ideal
Page 35 and 36: 1. N(I) if finite. 2. Norm is multi
Page 37 and 38: Let I be the non-zero integral idea

Abstract Algebra and Algebraic Number Theory

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