Math 135 Algebra
Spring 2005
Lecture Schedule
Note: The section numbers refer to the text book, An
Introduction to Mathematical Thinking: Algebra and Number Systems,
by Gilbert and Vanstone. The theorems listed here form the skeleton of
the lectures, and meant to help you focus on the more important parts
of the course.
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Week 1 (May 2 to 6):
Course logistics, and outline.
Sections 1.2 to 1.4: statements, connectives, truth
values, truth tables, equivalence of statements, proof of equivalence by
truth tables, and by logical reasoning, counterexamples to
equivalence; Quantifiers, truth value of quantified statements,
equivalence of quantified statements, proofs of equivalence and counterexamples
to non-equivalence.
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Week 2 (May 9 to 13):
Sections 1.5, 1.6, 4.1, 4.2: Negation of compound statements, and
simplification of negated statements, methods of proof for implications
and `iff' statements.
Principle of mathematical induction (4.13) , strong induction (4.18, a
more general version was done in class), recursive definitions,
proofs by induction.
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Week 3 (May 16 to 20):
Sections 2.1, 2.2: Properties of the divisibility relation (definition,
and Proposition 2.11), Division Algorithm (statement),
Greatest common divisor (definition), Euclidean Algorithm (Proposition
2.21, 2.22), Properties of GCD (Proposition 2.24 and its converse, 2.28),
extended Euclidean algorithm (2.25).
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Week 4 (May 25 to 27):
Sections 2.3, 2.5: Linear Diophantine equations and their complete
integer solution (Theorem 2.31), Prime numbers and their
properties (definition, Propositions 2.52, 2.54, 2.55, 2.57).
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Week 5 (May 30 to June 3):
Sections 2.5, 3.1, part of 3.2, 3.3: Properties of prime numbers (see
above),
Congruences and their properties (definition, 3.11, 3.12),
test for divisibility by 9 (3.21),
(After June 1) further properties of congruences (3.13, 3.14),
Fermat's little theorem (3.42).
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Week 6 (June 6 to 10):
Proof of Fermat's little theorem (3.42), solving
linear congruences (Theorem 3.54),
Chinese remainder theorem with two moduli (3.62), solving simultaneous
linear congruences (examples such as 3.63).
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Week 7 (June 13 to 17):
Chinese remainder theorem (3.62), Private Key Encryption (such as in
example 7.21), Public Key Encryption (7.3), The RSA scheme (7.42):
private and public key pairs, encryption and decryption, decryption
using Chinese remainder theorem.
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Week 8 (June 20 to 25):
Correctness of the RSA scheme (theorem 7.41), Fast exponentiation
(7.46), encryption using fast exponentiation, Complex numbers (8.1, 8.2):
roots of quadratic equations (8.11), addition, multiplication, inverse.
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Week 9 (June 27 to 29):
The Complex plane (8.3): geometric meaning of addition, modulus/absolute
value, conjugate,
Properties of complex numbers (8.25, 8.42, 8.44).
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Week 10 (July 4 to 8):
Polar representation (8.5, 8.53): conversion between standard and polar
representation, geometric meaning of multiplication,
DeMoivre's Theorem (8.61), ``exponential representation'' of complex
numbers, applications to trigonometric functions (Problem 9-124).
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Week 11 (July 11 to 15):
Exponential representation of complex numbers (8.6),
Roots of complex numbers (8.72), Fundamental theorem of Algebra (8.81,
without proof), Solving polynomial equations in the complex field
(examples solved in class), Modular arithmetic and finite fields (3.4):
addition, multiplication, inverse modulo an integer m.
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Week 12 (July 18 to 22):
The finite field Zp (where p is a prime),
solutions to polynomial equations in prime fields (examples solved in
class),
Polynomials and factoring (9.1): division by polynomials (Theorem 9.11),
Remainder Theorem (9.12), Factor Theorem (9.13), Theorem 9.17, factoring
polynomials over finite fields (9.9).
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Week 13 (July 25 to 29):
Polynomials over the complex numbers (9.2): theorem 9.21, conjugate
roots theorem 9.24, theorem 9.26,
Error-correction codes (lecture notes:
[ ps ]
[ pdf ]).