Introduction to number theory, TCD 2010/11
Syllabus
The ultimate goal of this course is to introduce the students to most of the basic concepts of number theory, at the same time demonstrating interactions of number theory with other areas of maths and giving an overview of number-theoretic methods and results of contemporary mathematics. This ambitious goal is achieved through combining rigorous proofs with only hints on proofs and even just vague ideas in some cases, the latter being more of a roadmap for future studies rather than an examinable material. The course will be accompanied by bi-weekly tutorials in the form of problem-solving sessions. The only prerequisites are basic linear algebra (vector spaces, dimensions) and group theory from the first year. Recommended reading consists of (selected chapters from) books [1,2,3,4,5,6] below.- Euclid's algorithm. Linear Diophantine equations and Frobenius's problem. Fundamental theorem of arithmetic.
- Infinitude of primes. Number theory meets analysis: Bertrand's postulate, more on distribution of primes, primes in arithmetic sequences.
- Modular arithmetic. Fermat's little theorem. Euler's theorem. Chinese Remainder Theorem. Quadratic residues. Quadratic reciprocity law.
- Number theory meets computer science and cryptography: the Agrawal--Kayal--Saxena primality test and the Rivest--Shamir--Adleman algorithm.
- Euler's totient function. Number theory meets combinatorics: Möbius inversion and its applications.
- Polynomials over a field. Gauss's lemma. Eisenstein's criterion. Dumas's criterion.
- Cyclotomic polynomials and applications: primes in the arithmetic sequence an=dn+1; Wedderburn's little theorem.
- Algebraic numbers. Liouville's theorem and examples of transcendental numbers.
- Number theory meets algebraic geometry: Pythagorean triples. More on Diophantine equations: n=4 case of Fermat's last theorem, Markov's equation etc.
- Fermat's last theorem for polynomials. What breaks for integers? (Mistakes of Cauchy and Lamé, Kummer's ideal numbers.) The abc-conjecture.
- Number theory meets topology: p-adic numbers, Ostrowski's theorem, Hensel's lemma and applications.
Materials
A sample exam paper [PDF]Solutions to the sample paper [PDF]
Homework 1, due on October 15 [PDF]
Homework 2, due on October 29 [PDF]
Homework 3, due on November 19 [PDF]
Homework 4, due on December 3 [PDF]
Homework 5, due on December 16 [PDF]
Tutorial 1, October 8 [PDF]
Tutorial 2, October 22 [PDF]
Tutorial 3, November 5 [PDF]
Tutorial 4, November 26 [PDF]
Tutorial 5, December 10 [PDF]
Handout "Two applications of cyclotomic polynomials" [PDF]
Bonus question 1 (the first person to solve it will get a 10% bonus added to their final exam mark):
Consider the n×n-matrix A with aij=gcd(i,j). Show that det(A)=φ(1)φ(2)...φ(n).
Update: I received two correct solutions to the first bonus question (by Jackson Aurisch and Colman Humphrey).
Bonus question 2 (the first person to solve it will get a 10% bonus added to their final exam mark):
Show that the sum of the infinite series
is transcendental.
Update: I received a correct solution to the second bonus question (by Colman Humphrey).
Textbooks
- H. Davenport, The higher arithmetic. Cambridge University Press, 2008. (Modular arithmetic, RSA etc.)
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Oxford University Press, 2008. (Many parts of the course are there; in particular, this book contains an awful lot of information on algebraic and transcendental numbers.)
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory. Springer, 1990. (Chinese remainder theorem, irreducibility, diophantine equations, modular arithmetic.)
- Neal Koblitz, P-adic numbers, p-adic analysis, and zeta-functions. Springer, 1984. (You would only need Chapter 1, on p-adic arithmetic.)
- Serge Lang, Math talks for undergraduates. Springer, 1999. (Especially talk 2, on Fermat theorem for polynomials, the abc-conjecture etc.)
- Victor V. Prasolov, Polynomials. Springer, 2004. (Your main focus should be Chapter 2, including various results on irreducibility.)