Intro Abstract Algebra
c 1997-8, Paul Garrett,
[email protected] http://www.math.umn.edu/~garrett/
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Contents (1) Basic Algebra of Polynomials (2) Induction and the Well-ordering Principle (3) Sets (4) Some counting principles (5) The Integers (6) Unique factorization into primes (7) (*) Prime Numbers (8) Sun Ze's Theorem (9) Good algorithm for exponentiation (10) Fermat's Little Theorem (11) Euler's Theorem, Primitive Roots, Exponents, Roots (12) (*) Public-Key Ciphers (13) (*) Pseudoprimes and Primality Tests (14) Vectors and matrices (15) Motions in two and three dimensions (16) Permutations and Symmetric Groups (17) Groups: Lagrange's Theorem, Euler's Theorem (18) Rings and Fields: de nitions and rst examples (19) Cyclotomic polynomials (20) Primitive roots (21) Group Homomorphisms (22) Cyclic Groups (23) (*) Carmichael numbers and witnesses (24) More on groups (25) Finite elds (26) Linear Congruences (27) Systems of Linear Congruences (28) Abstract Sun Ze Theorem (29) (*) The Hamiltonian Quaternions (30) More about rings (31) Tables 2
1. Basic Algebra of Polynomials Completing the square to solve a quadratic equation is perhaps the rst really good trick in elementary algebra. It depends upon appreciating the form of the square of the binomial x + y: (x + y)2 = x2 + xy + yx + y2 = x2 + 2xy + y2 Thus, running this backwards,
x2 + ax = x2 + 2( a2 )x = x2 + 2( a2 )x + ( a2 )2 , ( a2 )2 = (x + a2 )2 , ( a2 )2
Then for a 6= 0,
ax2 + bx + c = 0
can be rewritten as Thus,
0 = a0 = x2 + 2 2ba x + ac = (x + 2ba )2 + ac , ( 2ba )2 (x + 2ba )2 = ( 2ba )2 , ac
r
x + 2ba = ( 2ba )2 , ac r b x = , 2a ( 2ba )2 , ac
from which the usual Quadratic Formula is easily obtained. Fo