Download Number Theory

Number Theory
Sohail Bahmani
October 1, 2009
Carnegie Mellon
Outline
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Divisibility and Congruency
gcd and lcm
Prime Numbers
Chinese Remainder Theorem
Groups
Fields
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Divisibility and Congruency
 Def. We say integer b divides integer a if there
exist an integer c such that a = bc.
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Notation: b|a
For all integers n, 1|n
If c|a and c|b, then c|ax+by for all integers x and y
If b|a, then |b| ≤ |a|
If c|b and b|a, then c|a
If a|b and b|a, then |a| = |b|
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Divisibility and Congruency
 Division Algorithm: For every pair of integers a
and b, with b≠0, there are unique integers q and r
such that a = bq + r and 0 ≤ r < |b|.
 Def. For a non-zero integer n, we say integers a
and b are congruent modulo n if n|a-b
 Notation: a = b (mod n)
 If a1 = b1 (mod n) and a2 = b2 (mod n), then
 for all integers x and y, a1x + a2y = b1x + b2y (mod n)
 a1a2 = b1b2 (mod n)
 for all natural numbers m, a1m = b1m (mod n)
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gcd and lcm
 Def. We say d is a common divisor of non-zero
integers a1, a2,…, an if d|ai for i = 1,2,…,n
 Def. The greatest number among the common
divisors of non-zero integers a1, a2,…, an is called
the greatest common divisor of a1, a2,…, an (Is it
well-defined?)
 Notation: gcd(a1, a2,…, an)
 Def. The smallest number among the common
multiples of non-zero integers a1, a2,…, an is called
the least common multiple of a1, a2,…, an
 Notation: lcm(a1, a2,…, an)
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gcd and lcm
 Def. Integers a and b are said to be coprime (or
relatively prime) if gcd(a,b)=1
 gcd(a,b) = gcd(a,b+ka) for all integers k
 gcd(a,b) lcm(a,b) = ab
 If integers a and b are not both zero, ax+by=m
has a solution iff gcd(a,b)|m
 For integers a, b, and c
 gcd(a,bc)=1 iff gcd(a,b)=1 and gcd(a,c)=1
 if gcd(a,b)=1 and a|bc, then a|c
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Prime Numbers
 Def. We call a natural number p>1 a prime
number, if the only positive divisors of p are 1 and
p.
 Def. The natural numbers that are not prime are
called compound numbers.
 There are infinitely many prime numbers
 Fundamental Theorem of Arithmetic: - a.k.a.
Unique-Prime-Factorization Theorem - Every
natural number greater than 1 can be uniquely
written as a product of some prime numbers.
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Chinese Remainder Theorem
 Suppose m1, m2,…, mk are pairwise coprime. Then
the following system of congruency equations
have solution and all of the solutions are
congruent modulo m1m2…mk.
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x = a1 (mod m1)
x = a2 (mod m2)
…
x = ak (mod mk)
Proof: Let M be the product of mi’s and Mi = M/mi. If
Mi’Mi = 1 (mod mi) then x = M1’M1a1+…+ Mk’Mkak is a
solution.
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Groups
 Def. A set G together with a binary operator * is
called a group if
 For all a and b in G, a*b is also in G
[Closure]
 For all a, b and c in G, a*(b*c)=(a*b)*c [Associativity]
 There exist an element e in G such that a*e=e*a=a for
all a in G
[Identity Element]
 For all a in G, there exist element b in G such that
a*b=b*a=e [Inverse Element]
 Def. If for every a and b in group (G,*), a*b= b*a,
then G is an abelian (or commutative) Group
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Groups
 Examples:
 ({0,1},xor) is a group.
 Z and multiplication does not form a group.
 For natural number n, ({0,1,2,…,n-1},+n) is group
where +n is summation modulo n.
 Suppose m>1 and S is the set of natural numbers less
than m that are coprime to m. Then (S,×m) is a group
where ×m is multiplication modulo m.
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Groups
 Def. Under binary operator *, subset H of G is
said to be a subgroup if H and * form a group.
 Notation: H≤G
 Def. If H≤G, for every g in G the left coset and the
right coset of H containing g are defined as
gH={g*h| h in H} and Hg={h*g| h in H},
respectively.
 Def. The order of a finite group is the number of
elements in that group.
 Notation: |G|
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Groups
 Lagrange’s Theorem: Let G be a finite group and H is a
subgroup of G. Then the order of H divides the order of
G.
 Proof: Different left cosets of H are disjoint. Hence, G can be
represented as the union of disjoint left cosets of H.
Furthermore, the order of the left cosets of H is equal to the
order of H. Consequently, |H| divides |G|.
 Def. Euler’s phi function: For natural number n, φ(n)
denotes the number of natural numbers less than n that
are coprime to n.
 If p1r1p2r2…pkrk is the prime factorization of n then
φ(n)=(p1-1) (p2-1)…(pk-1)p1r1-1p2r2-1…pkrk-1.
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Groups
 Euler’s Theorem: For natural number n, if
gcd(a,n)=1 then aφ(n) = 1 (mod n).
 Proof: Suppose that d is the smallest natural number
such that ad = 1 (mod n) (why exist?). Using Lagrange
theorem we can show that d|φ(n). Thus, there exist
an integer k such that aφ(n) = adk = (ad)k = 1 (mod n).
 Fermat’s Little Theorem: ap = a (mod p) for prime
number p and integer a.
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Fields
 Def. Let + and * be two different binary operators
defined on a set F. Denote the identity elements
of + and * by 0, and 1, respectively. We call (F,+,*)
a field if
 (F,+) and (F\{0},*) are both abelian groups
 for a, b, and c in F, a*(b+c)=(a*b)+(a*c) [Distributivity]
 Examples:
 (Z,+,×) is not a field, it is rather a ring.
 (Q,+,×) is a field.
 (Zp,+p,×p) is a field [a Finite Field or Galois Field]
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Fields
 Def. Characteristic of a finite field is the smallest
number of 1’s that must be added to get 0.
 Theorem: Characteristic of any finite field is a
prime number
 Def. The order of a finite field is the number of its
elements.
 Theorem: The order of a finite field is a prime
power.
 Proof: See that F with characteristic p is a vector space
over Fp={1,1+1,…,1+1+…+1 (p times)} (i.e., the scalars
come from Fp).
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