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Units and Primes in Quadratic Fields

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Units and Primes in Quadratic Fields Units and Primes 1 / 20 Overview Evolution of Primality Norms, Units, and Primes Factorization as Products of Primes Units in a Quadratic Field 2 / 20 Rational Integer Primes Definition A rational integer m is prime if it is not 0 or ±1, and possesses no factors but ±1 and ±m. 3 / 20 Division Property of Rational Primes Theorem 1.3 Let p; a; b be rational integers. If p is prime and and p j ab, then p j a or p j b. 4 / 20 Gaussian Integer Primes Definition Let π; α; β be Gaussian integers. We say that prime if it is not 0, not a unit, and if in every factorization π = αβ, one of α or β is a unit. Note A Gaussian integer is a unit if there exists some Gaussian integer η such that η = 1. 5 / 20 Division Property of Gaussian Integer Primes Theorem 1.7 Let π; α; β be Gaussian integers. If π is prime and π j αβ, then π j α or π j β. 6 / 20 Algebraic Integers Definition An algebraic number is an algebraic integer if its minimal polynomial over Q has only rational integers as coefficients. Question How does the notion of primality extend to the algebraic integers? 7 / 20 Algebraic Integer Primes Let A denote the ring of all algebraic integers, let K = Q(θ) be an algebraic extension, and let R = A \ K. Given α; β 2 R, write α j β when there exists some γ 2 R with αγ = β. Definition Say that 2 R is a unit in K when there exists some η 2 R with η = 1. Definition Say that α 2 R is prime in K if α is not zero, not a unit, and whenever α = βγ for some β; γ 2 R, we have β or γ is a unit. 8 / 20 Warning The norm of an algebraic integer depends on the field in which we consider it. For example, for 3 2 Q, we have N(3) = 3. But for 3 2 Q(i), we have N(3) = 9. Norms of Algebraic Integers Let β 2 K = Q(θ), where θ is algebraic of degree n over Q. Let β1; : : : ; βn be the conjugates of β for K. Definition The norm of β for K is defined to be N(β) = β1 ··· βn: 9 / 20 Norms of Algebraic Integers Let β 2 K = Q(θ), where θ is algebraic of degree n over Q. Let β1; : : : ; βn be the conjugates of β for K. Definition The norm of β for K is defined to be N(β) = β1 ··· βn: Warning The norm of an algebraic integer depends on the field in which we consider it. For example, for 3 2 Q, we have N(3) = 3. But for 3 2 Q(i), we have N(3) = 9. 10 / 20 Integrality of Norms of Algebraic Integers Lemma 7.1 Let α be an algebraic integer in K. Then N(α) is a rational integer. 11 / 20 Multiplicativity of Norms of Products of Algebraic Integers Lemma 7.2 Let α; β be algebraic integers in K. Then N(αβ) = N(α)N(β): 12 / 20 Characterization of Algebraic Integer Units Lemma 7.3 Let α be an algebraic integer in K. Then α is a unit in K if and only if N(α) = ±1. 13 / 20 Norms and Primality Theorem 7.4 Let α be an algebraic integer in K. If N(α) is a rational prime, then α is prime in K. 14 / 20 Algebraic Integers in K can be Factored Let K = Q(θ) be a fixed algebraic number field. Theorem 7.5 Every nonzero, nonunit algebraic integer in K can be factored into a product of primes in K. Corollary 7.6 There are an infinite number of primes in K. 15 / 20 Units in Imaginary Quadratic Fields Theorem 7.7 p Let U be the set of units in a quadratic field Q( D) where D is a negative square-free rational integer. (i) If D = −1, then U = {±1; ±ig. p p n 1± −3 −1± −3 o (ii) If D = −3, then U = ±1; 2 ; 2 . (iii) Otherwise, U = {±1g. 16 / 20 Units in Real Quadratic Fields Theorem p Let U be the set of units in a quadratic field Q( D) where D is a positive square-free rational integer. Then there is a unique fundamental unit u > 1 for which U = {±uk : k 2 Zg. Note p p The fundamentalp unit inpQ( 2) is 1 + 2, and in Q( 3) it is 2 + 3. 17 / 20 p Determining Small Units in Q( 2) Lemma 7.8 p p There is no unit in Q( 2) between 1 and 1 + 2. Proof p p Suppose FTSOCp that = x + y 2 is a unit in Q( 2) with 1 < < 1 + 2. 2 2 By Lemma 7.3,p we have x − 2y = ±1. It follows that −1 < x − y 2 < 1. p Adding inequalities, we obtain that 0 < 2x < 2 + 2, and infer that x = 1. p p But then 1 < 1 + y 2 < 1 + 2, which cannot hold for any rational integer y. 18 / 20 p Determining All the Units in Q( 2) Theorem 7.9 p There are an infinite number of units in Q( 2). They are given by ±λn, where n 2 Z. Proof Idea p Let be a unit in Q( 2). p Note that since λ = 1 + 2 > 1, the intervals [λk ; λk+1) partition the positive real line as k ranges through Z. n n+1 Hence if > 0, then there is some n 2 Z with λ ≤p < λ . We then have λ−n is a unit with 1 ≤ λ−n < 1 + 2, and apply Lemma 7.8 to find that = λn. For < 0, we apply the argument above to the unit −. 19 / 20 Acknowledgement Statements of results follow the notation and wording of Pollard and Diamond's Theory of Algebraic Numbers. 20 / 20.
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