0. Introduction Math 407: Modern Algebra I Robert Campbell UMBC - - PowerPoint PPT Presentation

• 0. Introduction

Math 407: Modern Algebra I Robert Campbell

UMBC

January 29, 2008

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 1 / 22

Outline

1

Math 407: Abstract Algebra

2

Sources

3

Cast of Characters

4

Background Material

5

Applications & Follow-Up

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 2 / 22

UMBC Course Description

The basic abstract algebraic structures (rings, integral domains, division rings, fields and Boolean algebra) will be introduced, and the fundamental concepts of number theory will be examined from an algebraic perspective. This will be done by examining the construction of the natural numbers from the Peano postulates, the construction of the integers from the natural numbers, the rationals as the field of quotients of the integers, the reals as the ordered field completion of the rationals and the complex numbers as the algebraic completion of the reals. The basic concepts of number theory lead to modular arithmetic; ideals in rings; and to examples

• f integral domains, division rings and fields as quotient rings. The concept
• f primes yields the algebraic concepts of unique factorization domains,

Euclidean rings, and prime and maximal ideals of rings. Examples of symmetries in number theory and geometry lead to the concept of groups whose fundamental properties and applications will be explored.

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 3 / 22

Algebra

Def: Algebra is a branch of mathematics that utilizes symbols, as letters, to represent specific numbers, values of

• vectors. (Webster)

concerns the study of structure, relation and quantity. (Wikipedia) From the title “Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala” (The Compendious Book on Calculation by Completion and Balancing) by Al-Khwarizmi (circa 820 AD) Algebra generally refers to polynomial equations (aka algebraic equations)

Robert Campbell (UMBC)

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January 29, 2008 4 / 22

Abstract

Def: Abstract (adj) Thought of apart from concrete realities or specific objects. (vt) To summarize cow, cow, cow, dog, dog, dog − → 3 Loss, debt − → (−3) Hypotenuse of isosceles right triangle − → √ 2 (irrational) x2 + 1 = 0 − → i = √−1

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 5 / 22

Outline

1

Math 407: Abstract Algebra

2

Sources

3

Cast of Characters

4

Background Material

5

Applications & Follow-Up

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 6 / 22

Number Theory

aka Algebra over the integers

Solve 6x + 15y = 9 (Linear Diophantine equation) Solve x2 + y2 = z2 (Pythagorean triples) Solve x3 + 3y2 = 5 (Elliptic curve) Solve x3 + y3 = z3 (subcase of Fermat’s Last Theorem)

Robert Campbell (UMBC)

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January 29, 2008 7 / 22

Geometry

The sides of an isosceles right triangle are incommensurable ( √ 2 is irrational) [Pythagorus, ca 500 BC] Angles cannot be trisected (with compass and ruler) A right triangle whose sides are integers cannot have area which is a square or twice a square [Conj: Fermat, 165?, unproven]

Robert Campbell (UMBC)

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January 29, 2008 8 / 22

Algebra

Solutions to quadratic equations Solutions to cubic and quartic equations Can solutions to x5 + ax4 + 1 = 0 be expressed with just roots? Solutions to systems of linear equations Solutions to systems of polynomial equations

Robert Campbell (UMBC)

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January 29, 2008 9 / 22

Symmetries

Geometries: Groups of Isometries (distance preserving transformations) Symmetry Groups

Geometric Figures Tesselation & Crystallographic Groups

Topology

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 10 / 22

Outline

1

Math 407: Abstract Algebra

2

Sources

3

Cast of Characters

4

Background Material

5

Applications & Follow-Up

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 11 / 22

The Zoo

Groups Rings Fields Vector Spaces & Modules Algebras

Robert Campbell (UMBC)

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January 29, 2008 12 / 22

Groups

Def: A group is a set with a “multiplication” operation, an identity element (“multiplication” by it has no effect) and inverses. Integers with Addition: Z+ Modular Integers with Multiplication: Z∗

n

Matrix Groups with Multiplication: GLn(R), SL2(Z), etc Permutation Groups: Geometric Symmetries: Tesselations, Polygons, Polyhedra

Robert Campbell (UMBC)

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January 29, 2008 13 / 22

Rings

Def: A ring is a set with a “multiplication” operation and a commutative “addition” operation, an element which acts like “0” and another which acts like “1”, and additive inverses. Z, Q, R and C Square Matrices: Mn(R), Mn(Z), etc Polynomials: Z[x], C[x], Q[x, y], etc Modular Integers: Zn Algebraic Integers: Z[√−1], Z[ √ 3], etc Real Division Rings: R ⊂ C =< 1, i|i2 = −1 >⊂ H =< 1, i, j, k|i2 = j2 = k2 = ijk = −1 >

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 14 / 22

Fields

Def: A field is a set with a commutative “multiplication” operation and a commutative “addition” operation, an element which acts like “0” and another which acts like “1”, additive and multiplicative inverses (except for “0”). Q, R and C Number Fields: Q[√−1], Q[ √ 3], etc Finite Fields: Zp, GF(pn) := Zp[x]/ < p(x) >

Robert Campbell (UMBC)

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January 29, 2008 15 / 22

Vector Spaces & Modules

Def: A vector space V over a field F is a set with commutative addition and scalar multiplication by elements of the field. A module M over a ring R is a set with commutative addition and scalar multiplication by elements of the ring. Vector Space: Rn, Cn, Mn(R), R[x], etc Module: Zn, Zk1 ⊕ Zk2 ⊕ · · · , etc

Robert Campbell (UMBC)

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January 29, 2008 16 / 22

Algebras

Def: An algebra is a ring which is also a vector space. Square Matrices: Mn(R), etc Polynomials: C[x], Q[x, y], etc Real Division Algebras: R ⊂ C =< 1, i|i2 = −1 >⊂ H =< 1, i, j, k|i2 = j2 = k2 = ijk = −1 >

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 17 / 22

Outline

1

Math 407: Abstract Algebra

2

Sources

3

Cast of Characters

4

Background Material

5

Applications & Follow-Up

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 18 / 22

Linear Algebra

Matrix multiplication Non-commutative multiplication (AB = BA) Zero divisors (e.g. 1 1

• =
• )

Nilpotent elements (e.g. if N = 1

• then N2 = 0)

Idempotent elements (e.g. if A = 1

• then A2 = A, but A = 1)

Trace and Det of a linear transformation

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 19 / 22

Analysis

Logic Proofs Set Theory C and R - Construction and algebraic closure. Polynomial and rational functions

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 20 / 22

Outline

1

Math 407: Abstract Algebra

2

Sources

3

Cast of Characters

4

Background Material

5

Applications & Follow-Up

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 21 / 22

Further Topics

Abstract Algebra II (Math 408)

more Rings & Fields more Group Theory Galois Theory (combining Group & Field Theory)

Number Theory (Math 413) Algebraic Number Theory Algebraic Geometry Algebraic Topology Differential Geometry

Robert Campbell (UMBC)

• 0. Introduction

January 29, 2008 22 / 22