note: more information will be added as the exam dates near.

midterm exam: fri nov 3 (in class)

• answer true/false questions
• precisely state definitions/theorems with names (fundamental theorem of arithmetic, Lagrange's theorem, Fermat's little theorem, Euler's theorem, etc)
• some computational exercises (by hand, no calculators allowed)
• write one or two simple non-computational proofs (possibly of a result that was already proved in class, an exercise, or something new)

• determine if something is a group/ring/field
• compute the gcd of two numbers (possibly in a imaginary quadratic ring)
• use the extended Euclidean algorithm to find a solution (or prove none exist) to a linear Diophantine equation: ax + by = c (possibly over an imaginary quadratic ring)
• factor elements of irreducible quadratic rings into irreducibles (which includes determining if an element in a quadratic ring is irreducible)
• check that an element of a quadratic ring has two distinct factorizations or show why two factorizations are equivalent
• prove simple consequences of unique factorization (for Z or Z[i] or Z[sqrt(-2)])
• be able to use/prove basic divisibility tests
• do calculations (e.g., exponentiation by repeated squaring, or determination of squares, inverses, ...) mod n
• use modular arithmetic to show certain equations have no solutions, or possibly no solutions of a certain form
• carry out an example of RSA and/or prove why RSA works
• be familiar with the terms in the index (which occur up to p. 103)

final exam: fri dec 15 (1:30-3:30pm)

in terms of content, here are the main things you should be comforable with (this is not necessarily a comprehensive list of topics to study):

• all topics listed above for the midterms
• fermat's two square theorem, including various aspects of the proof (e.g., lagrange's lemma, the connection with unique factorization in Z[i], ...)
• understand the statement of the chinese remainder theorem and how to use it to determine the euler phi function, and what is say about lifting congruences mod m and mod n to congruences mod mn
• know the statement of and how to use quadratic reciprocity (including the first but not the second supplementary law) to compute legendre symbols and apply to diophantine equations such as x^2 + dy^2 = n (you do not need to know the proof of quadratic reciprocity)
• know the statements of the three and four square theorems, but not the proof or anything about quaternions
• know the structure theorem for units of real quadratic rings and the relation to Pell's equation (e.g., knowing how to compose solutions to Pell's equation)
• know how solutions of Pell's equation give good approximations to sqrt(d), including the "1/2y" error bound
• be able to compute fundamental units or +units for Z[sqrt(d)], using continued fractions or otherwise
• compute continued fractions of rational and quadratic numbers,
• be able to put together the above 4 bullet points to get rational approxiations to sqrt(d) within a specified bound, as in the final exercises of section 5.4 (section 5.5 will not be covered on the exam)

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