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KTH Royal Institute of Technology on Integers, Primes and Unique KTH Royal Institute of Technology Department of Mathematics Bachelor Thesis, SA104X On Integers, Primes and Unique Factorization in Quadratic Fields Author: Supervisor: Alice Hedenlund Dan Laksov May 18, 2013 1 Abstract. This thesis will deal with quadratic fields. The prob- lem is to study such fields and their properties including, but not limited to, determining integers, finding primes and deciding which quadratic fields have unique factorization. The goal is to get famil- iar with these concepts and to provide a starting point for students with an interest in algebra to explore field extensions and inte- gral closures in relation to elementary number theory. The reader will be assumed to have a basic knowledge in algebra and famil- iar with concepts such as groups, rings and fields. The necessary background material is covered in for example A First Course In Abstract Algebra by John B. Fraleigh. Some familiarity with basic number theory may be helpful, but not necessary for the scope of this thesis. The questions posed in this thesis was answered by means of literature and discussions with fellow students and my supervisor. The first four sections will deal with basic concepts in algebra such as algebraic numbers, algebraic integers and prime numbers. This knowledge will then be applied to the subject of quadratic fields. The thesis is concluded with two sections about important cases of quadratic fields, Gaussian and Eisenstein. 2 Contents 1. Algebraic Numbers 3 1.1. Algebraic Elements 3 1.2. Algebraic Extensions 4 2. Algebraic Integers 6 3. Prime Numbers 9 4. Factorization 10 4.1. UFDs and PIDs 10 4.2. Euclidean Domains 12 5. Quadratic Fields 13 5.1. Quadratic Integers 13 5.2. Unities and Primes 14 5.3. Unique Factorization in Quadratic Fields 15 6. Gaussian Integers 16 6.1. Basic Terminology 16 6.2. Gaussian Primes 17 6.3. Gaussian Integers as a Euclidean Domain 18 7. Eisenstein Integers 18 7.1. Basic Terminology 18 7.2. Eisenstein Primes 19 7.3. Eisenstein Integers as a Euclidean Domain 20 References 21 3 1. Algebraic Numbers In this first section we will explore some basic theory about algebraic numbers and algebraic field extensions. The section will not go into much detail, but focus on concepts that will help us in the upcomming sections. The reader looking for a more detailed study might want to refer the chapter "Algebraic Extensions" in Algebra by Serge Lang [5]. 1.1. Algebraic Elements. The concept of algebraic elements is some- thing that the reader should be familiar with, but as it is of such im- portance we will begin by a quick repetition of the definition. Definition 1.1. Let F be a subfield of E. An element α 2 E is said to be algebraic over F if it is a solution for some non-zero polynomial equation n n−1 anx + an−1x ··· + a1x + a0 = 0 for some ai 2 F and where n is a positive integer. If α is not algebraic over F , we call it transcendental over F . Definition 1.2. An element α 2 C is called an algebraic number if it is algebraic over Q. Similary, if α is transcendental over Q it is called an transcendental number. We continue with a couple of examples to illustate the difference between algebraic and transcendental numbers. p Example 1.3. The number 2 is an algebraic number since it is a zero of x2 − 2. In the same way, i is a root of x2 + 1, and is thus an algebraic number. Example 1.4. The real numbers π and e are examples of transcenden- tal numbers. However, the proof is not easy and beyond the scope of this thesis. For the full proof please refer to Introduction of the Theory of Numbers by G. Hardy and E. Wright [3]. We complement our terminology with a few qualities connected with algebraicity. Definition 1.5. Let E be a field extension of F and let α 2 E be algebraic over F . The polynomial with the smallest degree f(X) 2 F [X] such that f(α) = 0 is called the irreducible polynomial of α over F and is denoted Irr(α; F ). Observe that since F is a field we can easily normalize f(X) so that it has a leading coefficient equal to 1. We will later see that the degree of the irreducible polynomial is closely connected to the dimension of the vectorspace of E over F , which we denote [E : F ]. Definition 1.6. The degree of α over F , denoted deg(α; F ) is defined as the degree of Irr(α; F ). 4 p 1 Example 1.7. The element ρ = 2 (−1 + 3) has the irreducible poly- nomial X2 + X + 1 and is thus of degree 2. Example 1.8. The polynomial X4 − 2X2 + 2, which is irreducible by p p the Eisenstein critierion [2, p. 215] with p = 2, has the root 2 + 2 which is therefore an algebraic number of degree 4. Definition 1.9. Let E be a finite field extension of F . Let α 2 E and consider the F -linear function Tα(β) = αβ. We define the field norm as NE=F (α) = det Tα 1.2. Algebraic Extensions. We continue our study of algebraicity by looking into algebraic extensions. Algebraicity is an important prop- erty when it comes to field extensions, and algebraic extensions share a number of important qualities. We begin with a couple of definitions. Definition 1.10. Let F be a subfield of E. If every element of E is algebraic over F , we simply say that E is algebraic. Definition 1.11. We call K a number field if it is a finite extension of Q. We will continue with deriving a few propositions that will help us in the upcomming sections. Proposition 1.12. Let α be algebraic over F . Then the dimension [F (α): F ] is equal to deg(α; F ). Proof. Let deg(α; F ) = n. We need to show that the powers 1; α; : : : αn−1 are linearly independent over F. Assume the contrary. Then n−1 an−1α + ··· + a1α + a0 = 0 for some combination of ai 2 F , not all zero. Let n−1 f(X) = an−1X + ··· + a1X + a0 Since f(α) = 0 it must be true that Irr(α; F )jf(X) which is a contra- diction. Hence, the elements 1; α; : : : ; αn−1 form a base of F (α) over F and [F (α): F ] = n. Proposition 1.13. If E is finite as a vector space over F , then E is algebraic over F . Proof. Let [E : F ] = n and take an arbitrary element x 2 E. We know that any set of n + 1 elements in E are F -linearly dependent. Therefore the elements 1; x; x2; : : : ; xn are linearly dependant and we can find elements ai 2 F , not all zero, such that n n−1 anx + an−1x + ··· + a1x + a0 = 0 which shows that x is algebraic over F . Since x was arbitrary we can conclude that E is algebraic over F . 5 It is not true that all algebraic extensions are finite. An example is the set of algebraic numbers, which is an infinite extension of the rational numbers. Because of this, the set of algebraic numbers does not constitute a number field. Proposition 1.14. Let E be an extension field of F . The element α 2 E is algebraic over F if and only if F [α] = F (α). Proof. Let f(X) 2 F [X] be the smallest polynomial such that f(α) = 0 and let g(X) 2 F [X] be such that g(α) 6= 0. Then f(X) 6 j g(X) and by the Euclidean algorithm [1, p. 307] there are p(X) and q(X) such that f(X)p(X) + g(X)q(X) = 1 This means that g(α)q(α) = 1, from which can conclude that g(α) is invertible. Conversely, assume that F [α] = F (α). This means that α−1 2 F [α], i.e. −1 n α = a0 + a1α + ··· + anα for some ai 2 F . If we multiply with α we get 2 n+1 1 = a0α + a1α + ··· + anα substracting −1 from both sides shows us that α is algebraic. Specifically, this show us that inverses of algebraic numbers are alge- braic numbers themselves. Finally, we will conclude this chapter with showing that algebraic numbers are closed under addition and multi- plication. Proposition 1.15. If x and y are algebraic numbers, then x + y and xy are also algebraic numbers. Proof. Let x; y be algebraic numbers. Then m m−1 x = am−1x + ··· + a1x + a0 n n−1 y = bn−1y + ··· + b1y + b0 for some combination of ai 2 Q and bj 2 Q and for some positive integers m and n. Hence, we can view xm as a linear combination of xm−1; : : : ; x; 1. Consider a power of x greater than m, say xm+p. We want to show that this power also can be written as a linear combination of xm−1; : : : ; x; 1 for all positive integers p. We do this by induction on p. The case p = 0 is trivial. Assume that it is true for xm+p, then it suffices to show that it is also true for xm+p+1. Say x+p 0 m−1 0 0 x = am−1x + ··· + a1x + a0 Then for xm+p+1 m+p+1 m+p 0 m−1 0 0 x = xx = x(am−1x + ··· + a1x + a0) = 0 m 0 2 0 am−1x + ··· + a1x + a0x = 6 0 m−1 0 2 0 am−1 · (am−1x + ··· + a1x + a0) + ··· + a1x + a0x = 0 m−1 0 0 (am−1am−1)x + ··· + (am−1a1 + a0)x + am−1a0 Hence, xm+p+1 can also be written as a linear combination of xm−1; : : : ; x; 1, and we can conclude that all powers of x greater or equal to m can be written as a linear combination of xm−1; : : : ; x; 1.
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