Here are some Practice Materials for the final exam, from a course taught by another professor in 2011. Note that we have not discussed quadratic forms, so please omit items 16-21, 24-28 from the review sheet (along with the corresponding practice questions); omit also item 36; only partial answer is expected for 38, and you do not need to memorize it.
The first midterm covers sections 1.2-1.3, 2.1-2.3, 2.6-2.8, 2.10, 2.11 (with the exception of a few bits that were declared optional and not discussed in class.) See week-by-week schedule for more details. There will be 4 or 5 questions on the exam, mainly focusing on proofs but also including solving congruences. Easy applications of elementary congruences and divisibility criteria (as in Homework 2) are also included. The difficulty level will be similar to easier/medium questions on the homework. For practice, you can solve the textbook exercises in the corresponding sections: the book has a lot of questions, generally increasing in difficulty (the starred exercises are definitely above the exam level).
The first quiz is on Wednesday, 8/30, in class. The quiz will include two parts: 1) working from the definition, prove a basic divisibility property (similar to properties in Theorem 1.1 in the textbook); 2) apply the Euclidean algorithm to find the gcd of two given integers. (No calculators, the numbers will be small enough). You need to know how the Euclidean algorithm works and why.
Important: Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with other students in class, but your solutions should be written up in your own words, and all your collaborators should be listed. Please give complete proofs or explanations for all homework questions. No late homework is accepted.
Read the textbook! Optional but recommended: read Theorem 1.19 in Section 1.3 that gives another proof that there's infinitely many primes, by showing that the series Σ 1/p diverges.
Homework 1, due Wednesday, Sept 6: Section 1.2: 3ab, 11, 34, 35; Section 1.3: 11, 26 (6n+5 part only), 27, 42.
In questions from 1.2, please DO NOT use the prime decomposition (you can use it in questions from 1.3).
Hint for question 26: here's how you can solve a similar question, showing that there's infinitely many primes of the form (4n+3).
Optional challenge: if the homework seems too easy, find a question in section 1.2 or 1.3 that doesn't seem easy, and solve that question.
Week 2: Section 2.1 (beginning of the section only: basic properties of congruences). Applications: finding remainders, divisibility criteria for 2, 4, 5, 3, 9, properties of perfect squares (remainders mod 3 and mod 4 can only be 0 or 1). Here are some notes from the class. You should know and be able to prove properties like those of Theorem 2.1 and use them in elementary number theory problems such as in the homework. There will be a quiz on this material on Wednesday 9/13.
Week 3: Section 2.1: more congruences (Fermat's, Euler's, Wilson's theorem, applications), complete and reduced residue systems mod m. Read section 2.1 up to Lemma 2.13. Sections 2.10 and 2.11: additive group modulo m, multiplicative group modulo m. Fermat's and Euler's theorems via group theory. Important: everyone should know the basics of group theory, please review if necessary. Optional but recommended: read Lemmas 2.13, 2.14, and Theorem 2.15 in section 2.1.
Week 4: Sections 2.2, 2.3, 2.6. You should be able to solve congruences using the methods in these sections; know and use the Chinese Remainder theorem in various situations. In section 2.6, only theorem 2.23 and examples 11 and 12 are required. Theorem 2.24 and example 13 are optional. Sections 2.4 and 2.5 focus on numerical applications and are also optional.
Week 5: Sections 2.7, 2.8. Polynomial congruences. Modulo prime p, a congruence of degree n has at most n roots. Congruence x^d =1 mod p has exactly d roots if d divides (p-1). Order of a reduced residue class mod m, properties. Primitive roots, their properties; primitive roots as generators of cyclic multiplicative group mod m. Existence of primitive roots modulo p; examples showing that for some muduli m, there are no primitive roots. Using primitive roots and the group structure of R_{p} to solve congruence x^n = a mod p.
Read the textbook: we covered most of section 2.7 except the discussion at the very end. Read 2.8 up to Thm 2.39 (material starting from Theorem 2.39 is optional). Make sure you understand the argument in Theorem 2.37, Corollary 2.38, and subsequent discussion (alternative proof of Thm 2.12). You should be able to give complete solutions/proofs from scratch (NOT plugging in mumbers into Thm 2.37) for exercises 5, 6, 7, 9, 12 (do not submit any solutions, use these exercises for practice).
Homework 5, due MONDAY, OCT 2: Section 2.7: 4 (use the hint at the end of the book); Section 2.8: 13, 19, 22
Optional: exercise 36, section 2.8. This is a very long exercise, but it proves a pretty strong result. (None of this exercise is related to the exam; it is recommended as optional entertainment only.)
Midterm I will be on October 4, in class.
The exam covers sections 1.2-1.3, 2.1-2.3, 2.6-2.8, 2.10, 2.11 (with the exception of a few bits that were declared optional and not discussed in class.) See week-by-week schedule for more details. There will be 4 or 5 questions on the exam, mainly focusing on proofs but also including solving congruences. Easy applications of elementary congruences and divisibility criteria (as in Homework 2) are also included. The difficulty level will be similar to easier/medium questions on the homework. For practice, you can solve the textbook exercises in the corresponding sections: the book has a lot of questions, generally increasing in difficulty (the starred exercises are definitely above the exam level). An old exam (2011), by another professor. I recommend questions 1, 3, and 4; question 2 may require some of the material that we didn't cover.Week 7: Section 3.1 (up to Theorem 3.3); idea of quadratic reciprocity (statement of Theorem 3.4). We will finish 3.1 and 3.2 next week.
Homework 6, due Oct 18: Section 3.1: 9, 11, 13, 23 Section 2.8: 37. Hint for question 37: consider two cases, n even and n odd. For n odd, let p be the least prime divisor of n, and show that (p-1,n)=1. Then think about the order of 2 modulo p.
Week 8: Quadratic reciprocity, sections 3.1 and 3.2; you should know the definition and properties of the Legendre symbol, be able to compute it and use to determine whether a given class is a quadratic residue mod p or not. Representing integers as a sum of two squares (Fermat's theorem, end of section 2.1); you should know the statement and understand the ideas of the proof. Pythagorean triples, section 5.3; you should understand the proof. Diophantine equations, ad hoc methods, section 5.4; you should be able to use basic modular arithmetics arguments as well as the method of descent (as in Thm 5.10, but we'll use it in easier questions).
The exam covers only the material since the last midterm: sections 3.1, 3.2, 5.1- 5.4. Types of questions that will be on the test:
-- Compute the Legendre symbol, determine whether a given residue class is a quadradic residue modulo given p and whether the congruence x^2 =a mod p has solutions; find all odd primes p such that the given class is a quadratic residue modulo p;
-- Diophantine equations: 1) use basic divisibility and modular arithmetics, typically to show that the equation has no solutions; 2) method of descent; 3) be familiar with the ideas of proof for Pythagorean triples, use factorization to solve Diophantine equations. (You don't need to memoriza the formula for Pythagorean triples; if directly needed in the question, it will be provided.)
-- Linear Diophantine equations and systems.
Week 11: Sections 9.1, 9.2: polynomials, algebraic nummbers, algebraic integers. (We'll finish the proof of Theorem 9.12 next time). Please read the example of transcendental (not algebraic) number on p.418.Homework 9, due Nov 15: Section 9.1: 3, 5, 6. Section 9.2 : 1, 2. In question 5 section 9.1, use the ideas similar to Theorem 9.7.
Week 12: Section 9.3 (Thm 9.14 only), 9.4, 9.5.
Homework 10, due Nov 29: Section 9.4: 1, 2, 3. Section 9.5 : 1, 3, 4, 5.
Week 13: Sections 9.6 (Thm 9.22 only), 9.7, 9.8, part of 9.9 (Thm 9.28, Thm 9.29 parts 1-3 only, Example 1; please read Example 2 and work out the details).Homework 11, due Dec 6: Section 9.5: 6, Section 9.7: 1, 4, Section 9.9: 2, 3.
There will be a QUIZ on Monday, Dec 4. The quiz will cover polynomials (as in 9.1), algebraic numbers and algebraic integers (9.2; no need to memorize proofs of 9.11-9.12 but you need to know the statements); 9.4, 9.5 (for Thm 9.20, statement only), 9.6 (statement of Thm 9.22 only), 9.7. You need to know all the definitions and basic properties(algebraic numbers, algebraic integers, units, associates, primes, the norm in a quadratic field and its properties). I can ask you to prove some simple statements such as Thm 9.21 or Thm 9.24 or similar, and immediate applications. I will not ask you to do any lengthy calculations such as in the proofs of 9.22 or 9.27.
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