## Course Exams

### Exam 1: Wednesday September 30

The first exam will cover the material in the unstarred sections of Chapters 1, 2 and 3. You should be able to do the following
• State basic definitions (e.g., for divisibility, primes, groups, congruence, etc.).
• Prove the basic results we have shown in class. The most important ones are things with names, i.e., existence of prime factorization, prime divisor property, uniqueness of prime factorization, Lagrange's theorem, Fermat's little theorem, Wilson's theorem, and Lagrange's theorem on solutions of polynomials mod p.
• Do problems similar to those on Homeworks 1-5.

### Final Exam: Wednesday December 16, 4:30-6:30pm

The final exam will be cumulative, though it will mainly focus on Chapters 6, 7, and 9-12. In any case, you should be able to do the following
• State basic definitions: prime element, irreducible element, the norm (for elements of quadratic fields), unit, Legendre symbol, algebraic number/integer of degree m, ideal, divides (for elements or ideals), sum of ideals, product of ideals, prime ideal, maximal ideal, norm of an ideal, fractional ideal, class group, class number.
• State important results: Fermat's little theorem, Wilson's theorem, Lagrange's polynomial congruence theorem, the classification theorem for the solution of Pell's equation, Fermat's two square theorem, Lagrange's four square theorem, the (full) Chinese remainder theorem, quadratic reciprocity and the two supplementary laws, the determination of the rings of integers of (real or imaginary) quadratic fields, know which of Z, Z[i], Z[sqrt(-2)], Z[sqrt(-3)], Z[zeta_3] and Z[sqrt(-5)] have unique factorization/are PIDs, every PID has unique factorization, every maximal ideal is prime, the converse to these two for rings of integers, classification of prime and maximal ideals in terms of quotients, existence and uniqueness of prime ideal factorization. You should also understand the table in Section 11.7 of the notes.
• Compute: do modular arithmetic, the norm of elements of quadratic fields or ideals, Legendre symbols, determine if an element of a quadratic ring is a unit, irreducible, or prime, or factor it, Euler's phi function, determine a basis for a number field, determine if a number is an algebraic integer, sum of ideals, product of ideals, determine if an ideal is principal or not, determine a quotient ring, determine if an ideal is maximal or prime.
• Prove: determination of the primes of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+4y^2 or x^2+5y^2 (except for the first or fourth, I would tell you the precise statement), why Z, Z[i] or Z[zeta_3] has unique factorization (with or without ideals), why Z[sqrt(-3)] and Z[sqrt(-5)] don't have unique factorization, Euclidean doamins are PID's, PID's have unique factorization, prime ideals are maximal, the converse for rings of integers, the existence/uniqueness of factorization into prime ideals.

I will not ask you about the proof of quadratic reciprocity (or Euler's criterion or the Chinese remainder theorem, which are involved), since there are many proofs, and the one we gave is not among my favorites.

Note that many of the above results are fairly involved, so for the longer ones, I may ask you to either sketch the proof or prove one step of the result. If I ask you to sketch or explain why something is true, the more detailed of an explanation you can give me, the more points you will get (to a point).

For example, if I ask you to explain why Z[i] has unique factorization, simply stating that it is a PID will not get you many points. If you say it is a Euclidean domain, and therefore a PID, you'll get more points. If you briefly explain why it is a Euclidean domain first, you'll get full points. (Of course you could also explain this without using the notion of PID's, a we did in Chapter 6, but it's faster to quote the PID result.)

You should definitely expect that I will ask you to explain the determination of primes of the form x^2+ny^2 for at least one of n=1,2 or 3, as well as the n=5 case. You should also expect that I will ask you about the proof of existence and uniqueness of factorization in to prime ideals, as these ideas were the culmination of the course. Thus you should probably think about what answers you would give in advance, and if you have any questions about what the main points of the proof, or if I like your answers, then you should ask me in office hours beforehand.