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Integral Bases 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 2 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 2 K and choose r 2 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 2 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 2 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 2 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 2 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 2 Q(θ). Then N(β) 2 Q and Tr(β) 2 Q. If β is an algebraic integer, then N(β) 2 Z and Tr(β) 2 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 !1;:::;!n. Suppose FTSOC that !1;:::;!n is not an integral basis, and let ! be an algebraic integer which is not a Z-linear combination. We may suppose WLOG that ! = (a + r)!1 + a2!2 ··· + an!n where a 2 Z and 0 < r < 1. ∗ ∗ But then defining !1 = ! − a!1 and !i = !i for 2 ≤ i ≤ n, ∗ ∗ we deduce that !1;:::;!n 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. 14 / 14.
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