Journal articles: 'Algebraic number theory'

1

Blackmore, G. W., I. N. Stewart, and D. O. Tall. "Algebraic Number Theory." Mathematical Gazette 73, no. 463 (March 1989): 65. http://dx.doi.org/10.2307/3618234.

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2

S., R., and Michael E. Pohst. "Computational Algebraic Number Theory." Mathematics of Computation 64, no. 212 (October 1995): 1763. http://dx.doi.org/10.2307/2153389.

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3

Karve, Aneesh, and Sebastian Pauli. "GiANT: Graphical Algebraic Number Theory." Journal de Théorie des Nombres de Bordeaux 18, no. 3 (2006): 721–27. http://dx.doi.org/10.5802/jtnb.569.

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4

Lenstra Jr., H. W. "Algorithms in Algebraic Number Theory." Bulletin of the American Mathematical Society 26, no. 2 (October 1, 1992): 211–45. http://dx.doi.org/10.1090/s0273-0979-1992-00284-7.

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5

Platonov, V. P., and A. S. Rapinchuk. "Algebraic groups and number theory." Russian Mathematical Surveys 47, no. 2 (April 30, 1992): 133–61. http://dx.doi.org/10.1070/rm1992v047n02abeh000879.

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6

Appleby, Marcus, Steven Flammia, Gary McConnell, and Jon Yard. "SICs and Algebraic Number Theory." Foundations of Physics 47, no. 8 (April 24, 2017): 1042–59. http://dx.doi.org/10.1007/s10701-017-0090-7.

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7

Belabas, Karim. "Topics in computational algebraic number theory." Journal de Théorie des Nombres de Bordeaux 16, no. 1 (2004): 19–63. http://dx.doi.org/10.5802/jtnb.433.

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8

Schoof, Ren\'e. "Book Review: Algorithmic algebraic number theory." Bulletin of the American Mathematical Society 29, no. 1 (July 1, 1993): 111–14. http://dx.doi.org/10.1090/s0273-0979-1993-00392-6.

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9

Krishna, Amalendu, and Jinhyun Park. "Algebraic cobordism theory attached to algebraic equivalence." Journal of K-Theory 11, no. 1 (February 2013): 73–112. http://dx.doi.org/10.1017/is013001028jkt210.

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AbstractBased on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

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10

Cremona, J. E., and Henri Cohen. "A Course in Computational Algebraic Number Theory." Mathematical Gazette 78, no. 482 (July 1994): 221. http://dx.doi.org/10.2307/3618596.

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11

Wójcik, J. "On a problem in algebraic number theory." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (February 1996): 191–200. http://dx.doi.org/10.1017/s0305004100074090.

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Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’

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12

Lagarias, Jeffrey C., and Yang Wang. "Haar Bases forL2( ) and Algebraic Number Theory." Journal of Number Theory 76, no. 2 (June 1999): 330–36. http://dx.doi.org/10.1006/jnth.1998.2353.

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13

Bezushchak, O. O., and B. V. Oliynyk. "Algebraic theory of measure algebras." Reports of the National Academy of Sciences of Ukraine, no. 2 (May 3, 2023): 3–9. http://dx.doi.org/10.15407/dopovidi2023.02.003.

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A. Horn and A. Tarski initiated the abstract theory of measure algebras. Independently V. Sushchansky, B. Oliynyk and P. Cameron studied the direct limits of Hamming spaces. In the current paper, we introduce new examples of locally standard measure algebras and complete the classification of countable locally standard measure algebras. Countable unital locally standard measure algebras are in one-to-one correspondence with Steinitz numbers. Given a Steinitz number s such measure algebra is isomorphic to the Boolean algebra of s-periodic sequences of 0 and 1. Nonunital locally standard measure algebras are parametrized by pairs (s, r), where s is a Steinitz number and r is a real number greater or equal to 1. We also show that an arbitrary (not necessarily locally standard) measure algebra is embeddable in a metric ultraproduct of standard Hamming spaces. In other words, an arbitrary measure algebra is sofic.

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14

Aoki, Masao. "Deformation theory of algebraic stacks." Compositio Mathematica 141, no. 01 (December 1, 2004): 19–34. http://dx.doi.org/10.1112/s0010437x04000806.

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15

Thompson, Robert C. "Editorial announcement: Algebraic graph theory." Linear and Multilinear Algebra 28, no. 1-2 (October 1990): 1. http://dx.doi.org/10.1080/03081089008818025.

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16

Tamme, Georg. "Excision in algebraic -theory revisited." Compositio Mathematica 154, no. 9 (August 6, 2018): 1801–14. http://dx.doi.org/10.1112/s0010437x18007236.

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By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic$K$-theory. We give a new and direct proof of Suslin’s result based on an exact sequence of categories of perfect modules. In fact, we prove a more general descent result for a pullback square of ring spectra and any localizing invariant. Our descent theorem contains not only Suslin’s result, but also Nisnevich descent of algebraic$K$-theory for affine schemes as special cases. Moreover, the role of the Tor-unitality condition becomes very transparent.

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17

Mandal, Satya, and Yong Yang. "Intersection theory of algebraic obstructions." Journal of Pure and Applied Algebra 214, no. 12 (December 2010): 2279–93. http://dx.doi.org/10.1016/j.jpaa.2010.02.027.

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18

Levine, Marc. "Intersection theory in algebraic cobordism." Journal of Pure and Applied Algebra 221, no. 7 (July 2017): 1645–90. http://dx.doi.org/10.1016/j.jpaa.2016.12.022.

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19

Stengle, Gillbert. "Algebraic theory of differential inequalities." Communications in Algebra 19, no. 6 (January 1991): 1743–63. http://dx.doi.org/10.1080/00927879108824227.

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20

Perucca, Antonella, and Pietro Sgobba. "Kummer theory for number fields and the reductions of algebraic numbers." International Journal of Number Theory 15, no. 08 (August 19, 2019): 1617–33. http://dx.doi.org/10.1142/s179304211950091x.

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For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

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21

Dai, Shouxin, and Marc Levine. "Connective Algebraic K-theory." Journal of K-Theory 13, no. 1 (January 2, 2014): 9–56. http://dx.doi.org/10.1017/is013012007jkt249.

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AbstractWe examine the theory of connective algebraic K-theory, , defined by taking the −1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend to a bi-graded oriented duality theory when the base scheme is the spectrum of a field k of characteristic zero. The homology theory may be viewed as connective algebraic G-theory. We identify for X a finite type k-scheme with the image of in , where is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory of connective algebraic K-theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies with the universal oriented Borel-Moore homology theory having formal group law u + υ − βuυ with coefficient ring ℤ[β]. As an application, we show that every pure dimension d finite type k-scheme has a well-defined fundamental class [X]CK in ΩdCK(X), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.

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22

Baumslag, Gilbert, Alexei Myasnikov, and Vladimir Remeslennikov. "Algebraic Geometry over Groups I. Algebraic Sets and Ideal Theory." Journal of Algebra 219, no. 1 (September 1999): 16–79. http://dx.doi.org/10.1006/jabr.1999.7881.

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23

Aragona, J., A. R. G. Garcia, and S. O. Juriaans. "Algebraic theory of Colombeauʼs generalized numbers." Journal of Algebra 384 (June 2013): 194–211. http://dx.doi.org/10.1016/j.jalgebra.2013.03.005.

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24

van Hoeij, Mark, and Vivek Pal. "Isomorphisms of algebraic number fields." Journal de Théorie des Nombres de Bordeaux 24, no. 2 (2012): 293–305. http://dx.doi.org/10.5802/jtnb.797.

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25

Yuan, Pingzhi. "On algebraic approximations of certain algebraic numbers." Journal of Number Theory 102, no. 1 (September 2003): 1–10. http://dx.doi.org/10.1016/s0022-314x(03)00068-4.

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26

Kelly, Shane, and Matthew Morrow. "K-theory of valuation rings." Compositio Mathematica 157, no. 6 (May 20, 2021): 1121–42. http://dx.doi.org/10.1112/s0010437x21007119.

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We prove several results showing that the algebraic $K$-theory of valuation rings behaves as though such rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.

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27

Pezlar, Zdeněk. "Solving Diophantine Equations by Factoring in Number Fields." Journal of the ASB Society 2, no. 1 (December 27, 2021): 29–34. http://dx.doi.org/10.51337/jasb20211227004.

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In this text we provide an introduction to algebraic number theory and show its applications in solving certain difficult diophantine equations. We begin with a quick summary of the theory of quadratic residues, before diving into a select few areas of algebraic number theory. Our article is accompanied by a couple of worked problems and exercises for the reader to tackle on their own.

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28

Lagarias, Jeffrey C., and Yang Wang. "Haar Bases forL2(Rn) and Algebraic Number Theory." Journal of Number Theory 57, no. 1 (March 1996): 181–97. http://dx.doi.org/10.1006/jnth.1996.0042.

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29

Ozan, Yildiray. "On algebraic K-theory of real algebraic varieties with circle action." Journal of Pure and Applied Algebra 170, no. 2-3 (May 2002): 287–93. http://dx.doi.org/10.1016/s0022-4049(01)00129-3.

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30

Tanimoto, Ryuji. "Algebraic torus actions on affine algebraic surfaces." Journal of Algebra 285, no. 1 (March 2005): 73–97. http://dx.doi.org/10.1016/j.jalgebra.2004.10.021.

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31

Duflot, Jeanne, and C. Tyrel Marak. "A filtration in algebraic K-theory." Journal of Pure and Applied Algebra 151, no. 2 (July 2000): 135–62. http://dx.doi.org/10.1016/s0022-4049(99)00050-x.

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32

Dickmann, M., and F. Miraglia. "Algebraic K-theory of special groups." Journal of Pure and Applied Algebra 204, no. 1 (January 2006): 195–234. http://dx.doi.org/10.1016/j.jpaa.2005.04.002.

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33

Grayson, Daniel R. "Universal exactness in algebraic K-theory." Journal of Pure and Applied Algebra 36 (1985): 139–41. http://dx.doi.org/10.1016/0022-4049(85)90066-0.

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34

Shimakawa, Kazuhisa. "Multiple categories and algebraic K-theory." Journal of Pure and Applied Algebra 41 (1986): 285–304. http://dx.doi.org/10.1016/0022-4049(86)90114-3.

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35

Kida, Masanari. "Kummer theory for norm algebraic tori." Journal of Algebra 293, no. 2 (November 2005): 427–47. http://dx.doi.org/10.1016/j.jalgebra.2005.06.035.

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36

Perucca, Antonella, and Pietro Sgobba. "Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II." Uniform distribution theory 15, no. 1 (June 1, 2020): 75–92. http://dx.doi.org/10.2478/udt-2020-0004.

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AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

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37

Khanduja, Sudesh K. "The discriminant of compositum of algebraic number fields." International Journal of Number Theory 15, no. 02 (March 2019): 353–60. http://dx.doi.org/10.1142/s1793042119500167.

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For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

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38

Ensor, A. "Algebraic coalitions." algebra universalis 38, no. 1 (December 1997): 1–14. http://dx.doi.org/10.1007/s000120050035.

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39

Kramer, Linus, and Katrin Tent. "Algebraic Polygons." Journal of Algebra 182, no. 2 (June 1996): 435–47. http://dx.doi.org/10.1006/jabr.1996.0179.

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40

Guo, Xiao Qiang, and Zheng Jun He. "The Applications of Group Theory." Advanced Materials Research 430-432 (January 2012): 1265–68. http://dx.doi.org/10.4028/www.scientific.net/amr.430-432.1265.

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Since the classification of finite simple groups completed last century, the applications of group theory are more and more widely. We first introduce the connection of groups and symmetry. And then we respectively introduce the applications of group theory in polynomial equation, algebraic topology, algebraic geometry , cryptography, algebraic number theory, physics and chemistry.

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41

Abouzahra, M., and L. Lewin. "The polylogarithm in algebraic number fields." Journal of Number Theory 21, no. 2 (October 1985): 214–44. http://dx.doi.org/10.1016/0022-314x(85)90052-6.

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42

Rausch, U. "Character Sums in Algebraic Number Fields." Journal of Number Theory 46, no. 2 (February 1994): 179–95. http://dx.doi.org/10.1006/jnth.1994.1011.

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43

Nongmanee, Anak, and Sorasak Leeratanavalee. "Algebraic connections between Menger algebras and Menger hyperalgebras via regularity." Algebra and Discrete Mathematics 36, no. 1 (2023): 61–73. http://dx.doi.org/10.12958/adm2135.

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Menger hyperalgebras of rank n, where n is a fixed integer, can be regarded as a natural generalization of arbitrary semihypergroups. Based on this knowledge, an interesting question arises: what a generalization of regular semihypergroups is. In the article, we establish the notion of v-regular Menger hyperalgebras of rank n, which can be considered as an extension of regular semihypergroups. Furthermore, we study regularity of Menger hyperalgebras of rank n which are induced by some subsets of Menger algebras of rank n. In particular, we obtain sufficient conditions so that the Menger hyperalgebras of rank n are v-regular.

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44

Fischler, Stéphane. "Orbits under algebraic groups and logarithms of algebraic numbers." Acta Arithmetica 100, no. 2 (2001): 167–87. http://dx.doi.org/10.4064/aa100-2-4.

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45

Nishioka, Kumiko. "Algebraic independence of certain power series of algebraic numbers." Journal of Number Theory 23, no. 3 (July 1986): 354–64. http://dx.doi.org/10.1016/0022-314x(86)90080-6.

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46

Ben-Moshe, Shay, and Tomer M. Schlank. "Higher semiadditive algebraic K-theory and redshift." Compositio Mathematica 160, no. 2 (December 15, 2023): 237–87. http://dx.doi.org/10.1112/s0010437x23007595.

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We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $\mathrm {K}(n)$ - and $\mathrm {T}(n)$ -local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$ , then its semiadditive K-theory is of height $\leq n+1$ . Under further hypothesis on $R$ , which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$ , we show that its semiadditive algebraic K-theory is of height exactly $n+1$ . Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$ -localized K-theory, showing that they coincide for any $p$ -invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$ .

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47

Nauryzbayev, N. Zh, G. E. Taugynbayev, and M. Beisenbek. "Application aspects of algebraic number theory in financial mathematics." Bulletin of L.N. Gumilyov Eurasian National University. Mathematics. Computer Sciences. Mechanics series 130, no. 1 (2020): 93–102. http://dx.doi.org/10.32523/2616-7182/2020-130-1-93-102.

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48

Kleiner, Israel. "The Roots of Commutative Algebra in Algebraic Number Theory." Mathematics Magazine 68, no. 1 (February 1, 1995): 3. http://dx.doi.org/10.2307/2691370.

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49

Trotter, Hale. "Book Review: A course in computational algebraic number theory." Bulletin of the American Mathematical Society 31, no. 2 (October 1, 1994): 312–19. http://dx.doi.org/10.1090/s0273-0979-1994-00542-7.

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50

Kleiner, Israel. "The Roots of Commutative Algebra in Algebraic Number Theory." Mathematics Magazine 68, no. 1 (February 1995): 3–15. http://dx.doi.org/10.1080/0025570x.1995.11996267.

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Journal articles on the topic 'Algebraic number theory'

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