Algebraic Integers

1 / 9 Overview

Generalized Integers

Algebraic Integers

2 / 9 Desired Properties of Generalized Integers

Let Q(θ) be an algebraic number field. With the example of Z[i] as the “integers” in Q(i), we desire the following general properties of integers in Q(θ). (i) The integers form a ring, i.e., if α and β are integers in Q(θ), so are α + β, α − β, and αβ. (ii) If α is an integer in Q(θ) and is also a rational number, then α is a rational integer. (iii) If α is an integer, then so are the conjugates of α. (iv) If γ ∈ Q(θ), then nγ is an integer for some non-zero rational integer n.

3 / 9 Algebraic Integers

Definition An algebraic number is an algebraic integer if its minimal polynomial over Q has only rational integers as coefficients.

Example √ √ −1+ −3 Note that 7 is an algebraic integer, as is 2 . √ 1+ −5 On the other hand, 2 is not an algebraic integer.

4 / 9 Rational Numbers which are Algebraic Integers are Rational Integers

Proposition If α is a rational number which is also an algebraic integer, then α ∈ Z. Proof Idea a Write α = b where gcd (a, b) = 1 and b > 0. n X j Consider the minimal polynomial p(x) = cj x with cn = 1. j=0 a From p( b ) = 0, deduce that b = 1.

5 / 9 Roots of Monic Polynomials in Z[x] are Algebraic Integers

Lemma 6.1 If α satisfies any monic polynomial with rational integer coefficients then α is an algebraic integer. Proof Idea Let p(x) be the minimal polynomial for α over Q. Note that p(x) divides f (x) in Q[x]. ∗ ∗ ∗ ∗ Write f (x) = p(x)q(x) = cf p (x)q (x) where p (x), q (x) are primitive. From cf = 1, deduce that cp = 1.

6 / 9 Algebraic Integers form a Ring

Theorem 6.2 If Q(θ) is an algebraic number field, then the algebraic integers in it form a ring.

Corollary 6.3 The totality of algebraic integers forms a ring.

7 / 9 Some Nonzero Rational Integer Multiple of an Algebraic Number must be an Algebraic Integer

Theorem 6.5 If θ is an algebraic number, then there is a rational integer r 6= 0 such that rθ is an algebraic integer.

Proof Idea n X j Suppose θ is a root of f (x) = aj x ∈ Z[x] with an > 0. j=0 n n−1 X n−j−1 j Observe that θ is a root of an f (x) = aj an (anx) . j=0 Infer that anθ is a root of a monic polynomial.

8 / 9 Acknowledgement

Statements of results follow the notation and wording of Pollard and Diamond’s Theory of Algebraic Numbers.

9 / 9