Integral Bases

1 / 14 Overview

Integral Bases

Norms and Traces

Existence of Integral Bases

2 / 14 Basis

Let K = Q(θ) be an algebraic number ﬁeld of degree n. By Theorem 6.5, we may assume without loss of generality that θ is an algebraic integer. Deﬁnition A set of numbers β1, . . . βs in K is said to form a basis for K over F if for each element β in K there exists a unique set of numbers d1,..., ds such that

β = d1β1 + ··· + ds βs .

3 / 14 Integral Basis

Deﬁnition A set of algebraic integers α1, . . . , αs in K is called an integral basis of K if every algebraic integer γ in K can be written uniquely in the form

γ = b1α1 + ··· + bs αs

where b1,..., bs ∈ Z.

4 / 14 Integral Bases are Bases Lemma 6.7 An integral basis is a basis.

Proof Idea Let α1, ··· , αs be an integral basis of K. Let β ∈ K and choose r ∈ Z so that rβ is an algebraic integer. We may write

rβ = b1α1 + ··· + bs αs b b β = 1 α + ··· + s α r 1 r s

It remains to show that α1, ··· , αs is linearly independent over Q.

5 / 14 Conjugates for Q(θ)

Let θ be algebraic of degree n over Q, with conjugates θ1, . . . , θn over Q.

Pn−1 i Let β ∈ Q(θ), with β = i=0 ci θ = r(θ). Suppose that β has minimum polynomial g(x) of degree m over Q.

Deﬁnition The conjugates of β for Q(θ) are the numbers βi = r(θi ) for 1 ≤ i ≤ n. Note Observe that β has m conjugates over Q, but has n conjugates for Q(θ). By Theorem 5.7, we also have that m divides n.

6 / 14 Norms and Traces

Let β ∈ Q(θ), where θ is algebraic of degree n over Q.

Let β1, . . . , βn be the conjugates of β for Q(θ). Deﬁnition The norm of β for Q(θ) is deﬁned to be

N(β) = NQ(θ)(β) = β1 ··· βn.

The trace of β for Q(θ) is deﬁned to be

Tr(β) = TrQ(θ)(β) = β1 + ··· + βn.

7 / 14 Computing Norms and Traces

Let β ∈ Q(θ), where θ is algebraic of degree n over Q. m X j Let g(x) = bj x be the minimum polynomial for β over Q j=0 with roots β1, . . . , βm. Then

n/m n n/m N(β) = (β1 ··· βm) = (−1) b0 and n n Tr(β) = (β + ··· + β ) = − b . m 1 m m m−1

8 / 14 Multiplicativity of Norms and Additivity of Traces

Theorem Let β, γ ∈ Q(θ). Then N(βγ) = N(β)N(γ)

and Tr(β + γ) = Tr(β) + Tr(γ).

9 / 14 Integrality of Norms and Traces of Algebraic Integers

Theorem Let β ∈ Q(θ). Then N(β) ∈ Q and Tr(β) ∈ Q. If β is an algebraic integer, then N(β) ∈ Z and Tr(β) ∈ Z.

10 / 14 Integrality of Discriminants of Bases Consisting of Algebraic Integers

Lemma 6.8 If α1, . . . , αn is any basis of Q(θ) over Q consisting only of algebraic integers, then ∆[α1, . . . , αn] is a rational integer. Proof Idea This result follows from the last theorem since

i T i ∆[α1, . . . , αn] = det((αj ) )det((αj )) (1) (1) (2) (2) (n) (n) = det(αi αj + αi αj + ··· + αi αj )

= det(Tr(αi αj )).

11 / 14 Existence of Integral Bases Theorem 6.9 Every algebraic number ﬁeld has at least one integral basis.

Proof Idea From the collection of all bases for Q(θ) consisting of algebraic integers, choose one whose discriminant has smallest (nonzero) magnitude. Let its elements be ω1, . . . , ωn.

Suppose FTSOC that ω1, . . . , ωn is not an integral basis, and let ω be an algebraic integer which is not a Z-linear combination. We may suppose WLOG that ω = (a + r)ω1 + a2ω2 ··· + anωn where a ∈ Z and 0 < r < 1.

∗ ∗ But then deﬁning ω1 = ω − aω1 and ωi = ωi for 2 ≤ i ≤ n, ∗ ∗ we deduce that ω1, . . . , ωn is an integral basis with discriminant of smaller magnitude. 12 / 14 Invariance of Discriminants of Integral Bases

Theorem 6.10 All integral bases for a given algebraic number ﬁeld have the same discriminant.

13 / 14 Acknowledgement

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

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