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0. Introduction Math 407: Modern Algebra I Robert Campbell UMBC January 29, 2008 Robert Campbell (UMBC) 0. Introduction January 29, 2008 1 / 22 Outline Math 407: Abstract Algebra 1 Sources 2 Cast of Characters 3 Background Material

  1. 0. Introduction Math 407: Modern Algebra I Robert Campbell UMBC January 29, 2008 Robert Campbell (UMBC) 0. Introduction January 29, 2008 1 / 22

  2. Outline Math 407: Abstract Algebra 1 Sources 2 Cast of Characters 3 Background Material 4 Applications & Follow-Up 5 Robert Campbell (UMBC) 0. Introduction January 29, 2008 2 / 22

  3. 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 of integral domains, division rings and fields as quotient rings. The concept of 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

  4. 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) 0. Introduction January 29, 2008 4 / 22

  5. 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) → i = √− 1 x 2 + 1 = 0 − Robert Campbell (UMBC) 0. Introduction January 29, 2008 5 / 22

  6. Outline Math 407: Abstract Algebra 1 Sources 2 Cast of Characters 3 Background Material 4 Applications & Follow-Up 5 Robert Campbell (UMBC) 0. Introduction January 29, 2008 6 / 22

  7. Number Theory aka Algebra over the integers Solve 6 x + 15 y = 9 (Linear Diophantine equation) Solve x 2 + y 2 = z 2 (Pythagorean triples) Solve x 3 + 3 y 2 = 5 (Elliptic curve) Solve x 3 + y 3 = z 3 (subcase of Fermat’s Last Theorem) Robert Campbell (UMBC) 0. Introduction January 29, 2008 7 / 22

  8. 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) 0. Introduction January 29, 2008 8 / 22

  9. Algebra Solutions to quadratic equations Solutions to cubic and quartic equations Can solutions to x 5 + ax 4 + 1 = 0 be expressed with just roots? Solutions to systems of linear equations Solutions to systems of polynomial equations Robert Campbell (UMBC) 0. Introduction January 29, 2008 9 / 22

  10. 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

  11. Outline Math 407: Abstract Algebra 1 Sources 2 Cast of Characters 3 Background Material 4 Applications & Follow-Up 5 Robert Campbell (UMBC) 0. Introduction January 29, 2008 11 / 22

  12. The Zoo Groups Rings Fields Vector Spaces & Modules Algebras Robert Campbell (UMBC) 0. Introduction January 29, 2008 12 / 22

  13. 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: GL n ( R ), SL 2 ( Z ), etc Permutation Groups: Geometric Symmetries: Tesselations, Polygons, Polyhedra Robert Campbell (UMBC) 0. Introduction January 29, 2008 13 / 22

  14. 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: M n ( R ), M n ( Z ), etc Polynomials: Z [ x ], C [ x ], Q [ x , y ], etc Modular Integers: Z n Algebraic Integers: Z [ √− 1], Z [ √ 3], etc Real Division Rings: R ⊂ C = < 1 , i | i 2 = − 1 > ⊂ H = < 1 , i , j , k | i 2 = j 2 = k 2 = ijk = − 1 > Robert Campbell (UMBC) 0. Introduction January 29, 2008 14 / 22

  15. 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: Z p , GF ( p n ) := Z p [ x ] / < p ( x ) > Robert Campbell (UMBC) 0. Introduction January 29, 2008 15 / 22

  16. 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: R n , C n , M n ( R ), R [ x ], etc Module: Z n , Z k 1 ⊕ Z k 2 ⊕ · · · , etc Robert Campbell (UMBC) 0. Introduction January 29, 2008 16 / 22

  17. Algebras Def: An algebra is a ring which is also a vector space. Square Matrices: M n ( R ), etc Polynomials: C [ x ], Q [ x , y ], etc Real Division Algebras: R ⊂ C = < 1 , i | i 2 = − 1 > ⊂ H = < 1 , i , j , k | i 2 = j 2 = k 2 = ijk = − 1 > Robert Campbell (UMBC) 0. Introduction January 29, 2008 17 / 22

  18. Outline Math 407: Abstract Algebra 1 Sources 2 Cast of Characters 3 Background Material 4 Applications & Follow-Up 5 Robert Campbell (UMBC) 0. Introduction January 29, 2008 18 / 22

  19. Linear Algebra Matrix multiplication Non-commutative multiplication ( AB � = BA ) � 0 � � 1 � 0 1 0 � 0 � Zero divisors (e.g. = ) 0 0 0 0 0 0 � 0 1 � then N 2 = 0) Nilpotent elements (e.g. if N = 0 0 � 1 � 0 then A 2 = A , but A � = 1) Idempotent elements (e.g. if A = 0 0 Trace and Det of a linear transformation Robert Campbell (UMBC) 0. Introduction January 29, 2008 19 / 22

  20. 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

  21. Outline Math 407: Abstract Algebra 1 Sources 2 Cast of Characters 3 Background Material 4 Applications & Follow-Up 5 Robert Campbell (UMBC) 0. Introduction January 29, 2008 21 / 22

  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

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