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 | ab, then p | a or p | 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 | αβ, then π | α or π | 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 ∈ R, write α | β when there exists some γ ∈ R with αγ = β. Definition Say that  ∈ R is a unit in K when there exists some η ∈ R with η = 1.

Definition Say that α ∈ R is prime in K if α is not zero, not a unit, and whenever α = βγ for some β, γ ∈ 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 ∈ Q, we have N(3) = 3. But for 3 ∈ Q(i), we have N(3) = 9.

Norms of Algebraic Integers

Let β ∈ 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 ∈ 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 ∈ Q, we have N(3) = 3. But for 3 ∈ 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 √ 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, ±i}. √ √ n 1± −3 −1± −3 o (ii) If D = −3, then U = ±1, 2 , 2 . (iii) Otherwise, U = {±1}.

16 / 20 Units in Real Quadratic Fields

Theorem √ 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 ∈ Z}.

Note √ √ The fundamental√ unit in√Q( 2) is 1 + 2, and in Q( 3) it is 2 + 3.

17 / 20 √ Determining Small Units in Q( 2)

Lemma 7.8 √ √ There is no unit in Q( 2) between 1 and 1 + 2.

Proof √ √ Suppose FTSOC√ that  = x + y 2 is a unit in Q( 2) with 1 <  < 1 + 2.

2 2 By Lemma 7.3,√ we have x − 2y = ±1. It follows that −1 < x − y 2 < 1. √ Adding inequalities, we obtain that 0 < 2x < 2 + 2, and infer that x = 1. √ √ But then 1 < 1 + y 2 < 1 + 2, which cannot hold for any rational integer y.

18 / 20 √ Determining All the Units in Q( 2)

Theorem 7.9 √ There are an infinite number of units in Q( 2). They are given by ±λn, where n ∈ Z.

Proof Idea √ Let  be a unit in Q( 2). √ 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 ∈ Z with λ ≤√ < 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.

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