Math 413 Lecture 1

Introduction & Modern Algebra Review

30 Jan, 2001

1. Approaches to Number Theory

• Historical - The traditional approach. Driven by a sequence of problems, not always well linked. Well covered by NZM.
• Modern Algebraic - Provides a framework for understanding and unifying problems posed by preceding approaches. May conflict with historical approach as the natural order of topic differs. (We assume that students have already had introductory modern algebra class.)
• Computational - Driven by problems of recent interest to various applications, e.g. primality proof, factoring. My personal interest. Not well covered by NZM.

1.A. Historical Roots

1.B. Computational Problems

• Factoring large integers
• Finding large primes

2. The Cast of Characters

• N - Natural numbers: {1, 2, 3, ...} with + (a semigroup)
• Z - Integers (from the German zahlen): {..., -2, -1, 0, 1, 2, ...} with +, -, * (a ring)
• Q - Rationals: {p/q for integers p,q with non-zero q} with +, -, *, / (a field)
  e.g. 3/4, -725/1113
• R - Reals: (a field)
  e.g. sqrt(2) (algebraic), pi (transcendental), (3pi + sqrt(7))/2
• C - Complexes (algebraic closure of R, i.e. roots of all real coeff polys) {a+bi| a,b real} (a field)
• Z_N - Modular integers: (integers viewed as remainders of division by N) (a ring)
• GF(pⁿ) - Galois (finite) field of order pⁿ for p prime (a field)
• Q(z) - Algebraic number field (Polynomials of a fixed degree in z, where z is a root of some irreducible polynomial) (a field)

3. A Review of Modern Algebra

3.A. Groups

• Group
  • Defn: A group G is a set A with a single binary operation, * (a map AxA -> A carrying (a,b)->a*b), such that:
    1. Closure: If a and b are in A then a*b is also in A.
    2. Identity: There is a (unique) element denoted 1 in A such that for all a in A we have a*1=1*a =a
    3. Inverses: If a is in A then there is some b in A such that a*b = b*a = 1
    4. Associativity: (a*b)*c= a*(b*c)
  • Notation: The group operation is commonly written as multiplication, *, with the identity element denoted by 1. Sometimes the group operation is written as addition, +, with the identity element denoted by 0.
  • Examples:
    1. Z⁺: The integers with the addition operation
    2. Z₅⁺: The integers mod 5, together with the addition operation, i.e. [+,{0,1,2,3,4}]
    3. Z₅^*: The non-zero integers mod 5, together with the multiplication operation, i.e. [*,{1,2,3,4}]. Note that 1^(-1)=1, 2^(-1)=3 as 2*3=6=5+1=1 mod 5, 3^(-1)=2 and 4^(-1)=4
    4. Z₁₅^*: The integers relatively prime to 15 (i.e. not multiples of 3 or 5) mod 15, together with the multiplication operation, i.e. [*,{1,2,4,7,8,11,13,14}].
• Abelian Group
  • Defn: A group G is Abelian (or commutative) if, for any b and b in G we have a*b=b*a
  • Examples:
    1. Z_N^* is Abelian for any N
    2. M_n is group of (invertible) matrices is generally not Abelian.
    3. S₃: The group of permutations of three items (equiv the 6 rotations and flips of an equilateral triangle) is not Abelian as flip*rot is not equal to rot*flip.
• Generator
  • Defn: A group G is generated by a set of elements S = {a, b, ...} if every element of G can be written as a product of elements of S. The elements of S are called the generators of G.
  • Examples:
    1. Z₅^* is generated by the the set {2} as 2⁰=1 mod 5, 2¹=2 mod 5, 2²=4 mod 5 and 2³=3 mod 5
    2. Z₅^* is not generated by the set {4} as 4⁰=1 mod 5, 4¹=4 mod 5, but then higher powers repeat as 4²=1 mod 5 and 4³=4 mod 5
    3. Z₁₅^* is generated by the the set {2, 11} but not by any single element set.
• Cyclic Group
  • Defn: A cyclic group is a group which can be generated by a single element.
  • Notation: The cyclic (sub)group generated by a and consisting of the powers of a is denoted <a>.
  • Examples:
    1. Z₅^* is cyclic as it is generated by {2}
    2. Z₁₅^* is not cyclic as no set of fewer than two elements will generate it. The subgroup generated by a single element, for example <7> = {1, 7, 7²=49=4, 7³=13, 7³=13}, is cyclic.
• Order
  • Defn:
    1. The order of a group G is the number of distinct elements in the group.
    2. The order of an element a is the smallest positive exponent n such that aⁿ=1.
  • Note: The order of an element a is the order of <a>, the subgroup generated by a.
  • Notation: The order of a group is commonly denoted o(G). The order of an element is commonly denoted o(a).
  • Examples:
    1. Z₅^*: the group itself has order 4. Of the elements of Z₅^*, o(2)=4 and o(4)=2.
    2. Z₁₅^*: the group itself has order 8, i.e. o(Z₁₅^*) = 8. Of the elements of the group, o(2)=4 and o(11)=2.
• Direct Product/Sum
  • Defn: The sum of two groups G and H is a group whose elements are ordered pairs (g₁,h₁) and whose operation is defined by (g₁,h₁)*(g₁,h₁) = (g₁*_Gg₂, h₁*_Hh₂).
  • Note: The direct product and direct sum are the same thing if there is a finite number of terms. For an infinite number of terms the direct product and direct sum are different groups. The direct sum/product is also called the Abelian sum/product.
  • Notation: The sum of two groups is denoted G+H.
  • Examples:
    1. Z₁₅ = <2> + <11> = Z₄+Z₂.
• Quotient or Factor Group
  • Defn: The quotient of groups G and a (normal) subgroup H is a group whose elements are the cosets of G in H: {1H, g₁H, g₂H, ...} and whose operation is (g₁H)*(g₁H) = (g₁*_Gg₂H)
  • Notation: The quotient of G by H is commonly denoted G/H.
  • Examples:
    1. Z₅⁺ = Z⁺/5Z⁺, where Z₅⁺ has elements {0+5Z, 1+5Z, 2+5Z, 3+5Z, 4+5Z} or {{...,-5,0,5,...}, {...,-4,1,6,...}, {...,-3,2,7,...}, {...,-2,3,8,...}, {...,-1,4,9,...}}
• Lagrange's Theorem
  • Thm: If H is a subgroup of G then o(H) divides o(G)
  • Examples:
    1. Z₁₅^* has order 8. The subgroup <2> has order 4 and the subgroup <11> has order 2.

3.B. Rings

• Ring
  • Defn: A ring G is a set A with two binary operations, * and +, such that:
    1. Closure: If a and b are in A then both a*b and a+b are in A.
    2. Commutative Addition: a+b = b + a
    3. Associative Addition: a+(b+c) = (a+b)+c
    4. Additive Identity: There is a (unique) element denoted 0 in A such that for all a in A we have a+0=0+a =a
    5. Additive Inverses: If a is in A then there is some b in A such that a+b = b+a = 0
    6. Associative Multiplication: (a*b)*c = a*(b*c)
    7. Multiplicative Identity: There is a (unique) element denoted 1 in A such that for all a in A we have a*1=1*a =a
    8. Distributivity: a*(b)+c) = a*b +a*c and (a+b)*c = a*c +b*c)
  • Defn: A more concise way to define a ring is a set R with operations + and * and elements 0 and 1 such that [R,+,0] is an abelian group, [R,*,1] is a monoid (i.e. closure, associativity and identity) and the distributive laws hold.
  • Examples:
    1. Z - The integer (the original model for the ring concept)
    2. Q[x] - Polynomials with rational coefficients and usual addition and multiplication operations
    3. Z₁₅ - Integers mod 15 with the usual addition and multiplication operations
• Ideal
  • Defn: An ideal U of a ring R is an additive subgroup of R such that if u is in U and r is in R then both ur and ru are in U.
  • Notation: The ideal generated by elements a, b, ... is denoted <a, b, ...>. (This is a sloppy definition but adequate for the moment.)
  • Examples:
    1. 6Z = {..., -12, -6, 0, 6, 12, ...} is an ideal of Z
    2. <x+1, y+2> is an ideal of the two variable polynomial ring Z[x, y].

3.A. Fields

• Field
  • Defn: A field is a ring where multiplication is commutative and all elements but 0 have a multiplicative identity.
  • Note: Some references define non-commutative fields, but we will require multiplicative commutativity as part of the definition of a field.
  • Examples:
    1. C - The complex numbers
    2. Q - The rational numbers
    3. Z₅ - The integers mod 5 with usual addition and multiplication operations. Note that all elements other than 0 have a multiplicative inverse: 1^(-1)=1, 2^(-1)=3 as 2*3=6=5+1=1 mod 5, 3^(-1)=2 and 4^(-1)=4
    4. GF(4) - The quotient of the polynomial ring Z₂[x] by the ideal generated by (x²+x+1), so GF(4) = Z₂[x]/<x²+x+1>. The elements are represented by polynomials of degree 1 or less with coeffcients 0 and 1. Note that all elements other than 0 have inverses, as 1^(-1)=1, x^(-1)=x+1 as x(x+1) = x²+x = -1 = 1 and (x+1)^(-1)=x.
    5. Q[sqrt(-1)] - The Gaussian numbers (subfield of C). Contained in the Gaussian numbers is the ring Z[sqrt(-1)], the Gaussian Integers.
• Extension
  • Defn:
  • Examples: