Materials Covered and Suggested Readings, 1/24--2/18
Materials covered:
- Mathematical induction
- The fundamental theorem of arithmetic (primary factorization of
integers)
- Chinese remainder theorem
- Fermat's little theorem and Euler's theorem
- Every subset of integers closued under addition and multiplication
is the set of all multiples of an integer
- gcd and lcm
- Euclidean algorithm
- p-adic integers
- Hensel's Lemma
- The Mobius function
- Euler's phi function
- The "number of divisor" and "sum of divisors" function
- Existence of primitive elements modulo any prime number
Readings:
- gcd and lcm: section 3.3.
- Euclidean algorithm: section 3.4.
- The fundamental theorem of arithmetic: section 3.5.
- Chinese Remainder Theorem: section 4.2 of Rosen.
- Polynomial congruence equations and Hensel's Lemma,
in section 4.4 of Rosen.
(We will discuss it soon. But you might find it interesting to think
about how to solve a polynomial equation like x^2 - 3*y^2=1 modulo
higher and higher powers of a prime number p.
That is the content of Hensel's Lemma, analogues to Newton's method
for finding a root of an equation, starting with an approximate root.)
- Fermat's little theorem: section 6.1.
- Euler's theorem: section 6.3.
- Euler's phi function: section 7.1
- The sum and number of divisors function: section 7.2.
- The Mobius function: 7.4
- Existence of primitive elements modulo a prime number: sections 9.1-9.3.