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ExamplePaper_NumberTheoryCS290_S09

ExamplePaper_NumberTheoryCS290_S09 - QUANTUM ALGORITHMS IN...

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QUANTUM ALGORITHMS IN ALGEBRAIC NUMBER THEORY SIMON RUBINSTEIN-SALZEDO Abstract. In this article, we discuss some quantum algorithms for determining the group of units and the ideal class group of a number field. Assuming the generalized Riemann hypothesis, we will show furthermore that these algorithms require only quantum polynomial time. 1. Introduction Two very important problems in computational algebraic number theory are the com- putations of the unit group and the ideal class group of an algebraic number field. These groups are very important objects both in algebraic number theory and in other areas of mathematics. Ideal class groups of number fields were first studied by Gauß in 1798. They also played a major role in several early attempts at proving Fermat’s Last Theorem starting with the work of Kummer. In particular, if p is an odd prime and p does not divide the class number of Q ( ζ p ), where ζ p is a primitive p th root of unity, then it can be shown (see [5]) without too much di ffi culty that x p + y p = z p has no integer solutions in which p xyz . (The case of p | xyz is also treated in [5], but it is more di ffi cult.) In this paper, we discuss algorithms that compute the unit group and the ideal class group of a number field in quantum polynomial time. The algorithms we study here are due to Hallgren [2]. In the classical case, these two problems are typically solved simultaneously. In the quantum case, however, we first need to compute the unit group, and then we use the result of that computation to compute the ideal class group. 2. Number Theoretic Preliminaries Of key importance in algebraic number theory is the Galois group of a field extension; if L/K is a field extension, then Gal( L/K ) is the group of field automorphisms of L that fix every element of K . We frequently write elements of the Galois group multiplicatively, i.e. we write x σ rather than σ ( x ). Definition 1. An (algebraic) number field K is a finite field extension of the field of rational numbers Q contained in the field of complex numbers C . The ring of integers o = o K of K is the set of roots of monic polynomials f ( x ) Z [ x ] lying in K . The degree of K is the dimension of K considered as a vector space over Q ; we write [ K : Q ] for this number. If [ K : Q ] = d , and o = Z α 1 + · · · + Z α d , then the discriminant of K is defined to be Δ = det(Tr( α i α j )) 1 i,j, d , where Tr : F Q is given by x σ Gal( K/ Q ) x σ . 1
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2 SIMON RUBINSTEIN-SALZEDO It turns out (see e.g. Chapter 4 of [1]) that the structure of the group of units o × of o can be described quite explicitly. Theorem 2. (Dirichlet’s Unit Theorem.) Suppose K is a number field. If K has r 1 distinct embeddings into R and r 2 complex conjugate pairs of embeddings into C , and μ K is the group of roots of unity of K , then o × = μ K Z r 1 + r 2 - 1 .
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