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Mak Trifkovic - Algebraic Theory of Quadratic Numbers

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Mak Trifkovic Algebraic Theory of Quadratic Numbers
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By focusing on quadratic numbers, this advanced undergraduate or masters level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group. The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms. The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes.

The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields. The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders. Prerequisites include elementary number theory and a basic familiarity with ring theory.

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Mak Trifkovic - Algebraic Theory of Quadratic Numbers
Algebraic Theory of Quadratic Numbers
Mak Trifkovic
Algebraic Theory of Quadratic Numbers

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Mak Trifkovi Universitext Algebraic Theory of Quadratic Numbers 2013 10.1007/978-1-4614-7717-4_1 Springer Science+Business Media New York 2013
1. Examples
Mak Trifkovi 1
(1)
Department of Math and Statistics, University of Victoria, Victoria, BC, Canada
Abstract
When can we express a prime number as a sum of two squares? Lets start by sorting the first dozen primes into those with such an expression, and the rest:
Do you see a pattern 11 Review of Elementary Number Theory When can we - photo 1
Do you see a pattern?
1.1 Review of Elementary Number Theory
When can we express a prime number as a sum of two squares? Lets start by sorting the first dozen primes into those with such an expression, and the rest:
Do you see a pattern This question was posed by Pierre de Fermat a French - photo 2
Do you see a pattern?
This question was posed by Pierre de Fermat, a French eighteenth-century mathematician. It may strike you as an unmotivated riddle, like his more famous Last Theorem. One of the joys of number theory is that even such riddles often lead to beautiful, intrinsically interesting discoveries. Fermats question is our first example of the need to study divisibility, primes, and factorizations in rings bigger than Picture 3 .
In Ch. 1 we will look at concrete examples of such higher arithmetic. To understand them you only need to know the definition of a ring and a field. It helps to have some knowledge of ideals, but we leave an in-depth study of those for the next chapter. By convention, all our rings are commutative and have a multiplicative identity.
Recall that an integer p 0,1 is prime when it has no integer divisors other than1 and p . If a prime p is a sum of two squares, the equalities
show that p acquires a nontrivial factorization once we allow factors from the - photo 4
show that p acquires a nontrivial factorization once we allow factors from the set
Like this is a ring a set closed under addition subtraction and - photo 5
Like Picture 6 , this is a ring: a set closed under addition, subtraction, and multiplication, but not necessarily division. It is in this new ring that we need to answer the basic arithmetical question, how does p factor? The goal of the book is to rigorously pose and answer such questions in quadratic rings, like Algebraic Theory of Quadratic Numbers - image 7 . Those rings result from enlarging Algebraic Theory of Quadratic Numbers - image 8 by a solution to a quadratic equation (in our example, Algebraic Theory of Quadratic Numbers - image 9 ). General algebraic number theory undertakes the same task for polynomial equations of any degree. Most of its features, however, are already present in the quadratic case.
The main result of elementary arithmetic is the following well-known theorem.
1.1.1 Theorem (Unique Factorization in Picture 10 ).
Any integer other than 0 and 1 can be written as a product of primes, uniquely up to permuting the prime factors and changing their signs.
The proof of Thm. 1.1.1 will be our template for extending arithmetic to rings such as Algebraic Theory of Quadratic Numbers - image 11 . It proceeds through a chain of propositions.
1.1.2 Proposition (Division Algorithm).
Given Algebraic Theory of Quadratic Numbers - image 12 , there exist unique such that Proof We prove the existence of q and r leaving uniqueness - photo 13 such that
Proof We prove the existence of q and r leaving uniqueness as an exercise - photo 14
Proof.
We prove the existence of q and r , leaving uniqueness as an exercise. Lets first assume that b >0. Check that the set
is nonempty By the Well-Ordering Principle has a minimal element r which - photo 15
is nonempty. By the Well-Ordering Principle, Picture 16 has a minimal element r , which must be of the form for some If we had b r then so that r b S We also have r b lt r - photo 17 for some If we had b r then so that r b S We also have r b lt r since b - photo 18 . If we had b r , then so that r b S We also have r b lt r since b gt0 which contradicts - photo 19 , so that r b S . We also have r b < r (since b >0), which contradicts the choice of r as the minimum of S .
If b <0, we apply the preceding reasoning to a and b to find Algebraic Theory of Quadratic Numbers - image 20 with Algebraic Theory of Quadratic Numbers - image 21 . We then put Algebraic Theory of Quadratic Numbers - image 22 .
Let Picture 23 , not both zero. The term greatest common divisor (g.c.d.) of a and b, denoted gcd( a , b ), is self-explanatory. The next link in our chain of reasoning describes the important properties of the g.c.d.
1.1.3 Proposition (Euclids Algorithm).
Let a and b be in not both zero The greatest common divisor of a and b satisfies the following - photo 24 , not both zero. The greatest common divisor of a and b satisfies the following condition:
Algebraic Theory of Quadratic Numbers - image 25
(1.1.4)
Moreover, there exist Algebraic Theory of Quadratic Numbers - image 26 such that Algebraic Theory of Quadratic Numbers - image 27 .
Proof.
We prove the Proposition in a roundabout way that foreshadows the techniques we will use later. Since we want to express gcd( a , b ) as a linear combination of a and b with integer coefficients, we consider the set of all such combinations,
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