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Gaussian Integers and Other Quadratic Integer Rings

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DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2021 Gaussian Integers and Other Quadratic Integer Rings ERIK LANDIN SEIF HUSSEIN KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Abstract This thesis deals with quadratic integer rings, in particular the Gaus- sian integers Z[i]. Concepts such as quadratic extensions, Euclidean do- mains and unique factorization domains will be introduced to the reader. The goal of this thesis is to show how a natural generalization of the integers Z, in the form of the Gaussian integers, can be used to prove important results in Z. Furthermore, it aims to explore other types of quadratic integer rings and their differences. The first two sections will introduce the Gaussian integers, integral do- mains and norms. In the third section the irreducible elements of the Gaussian integers are categorized. The fourth and fifth sections talk about general quadratic integer rings and their norms. Section six and seven cat- egorizes the prime elements and introduces unique factorization domains. Sammanfattning Den h¨aruppsatsen avser kvadratiska heltalsringar, och s¨arskiltde Gaussiska heltalen Z[i]. Koncept som kvadratiska utvidgningar, Euklidisk dom¨anoch unik faktoriseringsdom¨anintroduceras till l¨asaren.M˚aletmed uppsatsen ¨aratt visa hur en naturlig generalisering av heltalen Z, i form av de Gaussiska heltalen, kan anv¨andasf¨oratt bevisa viktiga resultat i Z. Dessutom avser den att utforska olika typer av kvadratiska heltalsringar och deras skillnader. De tv˚af¨orstasektionerna kommer att introducera de Gaussiska heltalen, heltalsdom¨aneroch normer. I den tredje sektionen kategoriseras de irre- ducibla elementen hos de Gaussiska heltalen. Den fj¨ardeoch femte sektio- nen n¨amnerallm¨anakvadratiska heltalsringar och deras normer. Sektion sex och sju kategoriserar primtalselement och introducerar begreppet unik faktoriseringsdom¨an. 1 Acknowledgements We would like to thank our supervisor Roy Skjelnes for the guidance and help that he has offered us in writing this thesis. 2 Contents 1 Introducing complex numbers and the Gaussian integers 4 1.1 Identifying the Gaussian integers with matrices . 4 1.2 Gaussian integers as a ring . 5 1.3 Inverse . 6 1.4 Gaussian integers as an Integral domain . 7 2 Norm of the Gaussian integers 8 2.1 Units of the Gaussian integers . 8 3 Irreducible elements of the Gaussian integers 9 4 Quadratic extensions 13 4.1 Integral closure of a subring . 13 4.2 Integral elements of a quadratic extension . 13 5 Norm of quadratic integer rings 16 5.1 The field norm . 16 5.2 Units of quadratic integer rings . 18 6 Euclidean domains 21 6.1 Gaussian integers as a Euclidean domain . 21 6.2 Prime elements . 22 7 Principal ideal domains and unique factorization domains 23 7.1 Principal ideal domains . 23 7.2 Unique factorization domains . 23 7.3 Integers D that result in a UFD . 25 3 1 Introducing complex numbers and the Gaus- sian integers The complex numbers, C, is defined as the set numbers of the form a+bi, where a and b are real numbers. They have addition (a+bi)+(c+di) = (a+c)+(b+d)i and multiplication (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The Gaussian integers Z[i] are the integer lattice of the complex numbers a+bi, meaning a and b are integers. Figure 1.1: The integer lattice Z[i]. 1.1 Identifying the Gaussian integers with matrices We will begin by looking at the subset of real 2 × 2-matrices of the form a −b . b a This subset is certainly closed under addition. Also under multiplication since a −b c −d ac − bd −(ad + bc) = . Multiplication is moreover com- b a d c ad + bc ac − bd ac − bd −(ad + bc) c −d a −b mutative since = . Call this subset ad + bc ac − bd d c b a R2×2, we will show that it can be identified with C. To show this we need a bijection φ between R2×2 and C. a −b Let φ: −! be the function that sends z = a + bi to the matrix . C R2×2 b a To show it is bijective, we note that φ(a + bi) = φ(c + di) corresponds to an a −b c −d equality between matrices = . So it is injective and for each b a d c 4 a −b we can find a corresponding element in , simply choose a + bi. We b a C also need it to preserve the two operations. To show this note that, a + c −(b + d) φ((a + bi) + (c + di)) = = φ(a + bi) + φ(c + di) b + d a + c meaning it is preserves addition. Also note that, ac − bd −(ad + bc) φ((a + bi)(c + di)) = φ((ac − bd) + (ad + bc)i) = ad + bc ac + bd a −b c −d = = φ(a + bi)φ(c + di) b a d c therefore also preserving multiplication. We now have an isomorphism between R2×2 and C. Hence, results for these matrices when a and b are integers will also apply to the Gaussian integers. 1.2 Gaussian integers as a ring We will now present the algebraic structure known as a ring. Definition 1.1. A ring R is a nonempty set where we require R to have two binary operations [+; ·], which we will refer to as addition and multiplication respectively. Those binary operations must fulfill the following conditions: • Commutativity with respect to addition meaning a+b = b+a for a; b 2 R. • Associativity with respect to addition, meaning (a + b) + c = a + (b + c) for a; b; c 2 R. • Associativity with respect to multiplication, meaning (ab)c = a(bc) for a; b; c 2 R. • The existence of an element 0 such that a + 0 = 0 + a = a for all a 2 R, and 1 2 R. • An additive inverse, meaning for every element a 2 R, there exists an element −a in R such that a + (−a) = 0. • Two-sided distributivity with respect to multiplication, meaning for any a; b; c 2 R a(b + c) = ab + ac (a + b)c = ac + bc Any subset of a ring that fulfills these conditions is called a subring. If the multiplication operation is also commutative, a · b = b · a for all a; b 2 R, the ring is said to be a commutative ring. An element u of R is called a unit in R if there is some v in R such that vu = 1 and uv = 1. 5 Example. The Gaussian integers is a commutative ring. The integer lattice Z[i] is clearly an Abelian group under addition. As for multiplication it is com- mutative since C is commutative and Z[i] is a subring of C and it is associative since ((a+bi)(c+di))(e+fi) = (a+bi)((c+di)(e+fi)). We also have distribu- tivity since (a+bi)((c+e)+(d+f)i) = a((c+e)+(d+f)i)+b((c+e)i−(d+f)) = (a + bi)(c + di) + (a + bi)(e + fi). Therefore, the criteria for Z[i] to be a com- mutative ring are fulfilled. 1.3 Inverse Investigating the multiplicative inverse in the complex plane. We may assume a−bi that not both a and b in a+bi are zero. Looking at the product (a+bi) a2+b2 = 1, a−bi we conclude that the multiplicative inverse in C is a2+b2 . In Z[i] each element a−bi will not always have an inverse since a2+b2 is not always a Gaussian integer. 6 1.4 Gaussian integers as an Integral domain Here we introduce the concept of an Integral domain. Definition 1.2. An Integral domain is a commutative ring R such that ab 6= 0 for all nonzero a; b 2 R. Example. We have shown that the Gaussian integers are a commutative ring. a −b Since the Gaussian integers are isomorphic to and the determinant b a of this matrix is a2 + b2, then the determinant is greater than zero as long as a; b 6= 0. It follows that the product of two matrices of this type has a determinant greater than zero since the determinant is multiplicative, meaning the Gaussian integers make up an integral domain. Definition 1.3. Let R be an integral domain. • Suppose r 2 R is nonzero and is not a unit. Then r is called irreducible in R if whenever r = ab with a; b 2 R, at least one of a or b must be a unit in R. Otherwise, r is said to be reducible. • A nonzero element p is a prime if it is not a unit and whenever p divides ab for any a; b 2 R, then either p divides a or p divides b. If r 2 R and r = ab where a; b 2 R and a is a unit, then we say that r is equal to b up to associates. 7 2 Norm of the Gaussian integers This section will introduce a norm on the Gaussian integers. Definition 2.1. A function N : R −! N is a norm on the integral domain R. If N(a) > 0 for a 6= 0 then define N to be a positive norm. Let N : Z[i] −! N be N(a + bi) = a2 + b2. Then N is a positive norm on the Gaussian integers a2 + b2 > 0 if not a; b = 0. This is the same thing a −b as taking the determinant of the corresponding matrix of the form , b a a −b since det( ) = a2 + b2. Letz ¯ denote the complex conjugate of z, that is b a z¯ = a +¯ bi = a−bi, then we also have that N(z) = z·z¯ = (a+bi)(a−bi) = a2+b2.
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