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Journal articles on the topic 'Galois closure'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Bastianelli, F., G. P. Pirola, and L. Stoppino. "Galois closure and Lagrangian varieties." Advances in Mathematics 225, no. 6 (December 2010): 3463–501. http://dx.doi.org/10.1016/j.aim.2010.06.006.

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2

Antei, Marco, and Michel Emsalem. "Galois closure of essentially finite morphisms." Journal of Pure and Applied Algebra 215, no. 11 (November 2011): 2567–85. http://dx.doi.org/10.1016/j.jpaa.2011.02.019.

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3

Valentini, Robert C. "Quintic polynomials having Galois closure of genus zero with Galois group A5." Finite Fields and Their Applications 59 (September 2019): 97–103. http://dx.doi.org/10.1016/j.ffa.2019.05.004.

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4

Endo, Shizuo. "The rationality problem for norm one tori." Nagoya Mathematical Journal 202 (June 2011): 83–106. http://dx.doi.org/10.1017/s0027763000010266.

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AbstractWe consider the problem of whether the norm one torus defined by a finite separable field extensionK/kis stably (or retract) rational overk. This has already been solved for the case whereK/kis a Galois extension. In this paper, we solve the problem for the case whereK/kis a non-Galois extension such that the Galois group of the Galois closure ofK/kis nilpotent or metacyclic.

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5

Pinus, A. G. "The classical Galois closure for universal algebras." Russian Mathematics 58, no. 2 (February 2014): 39–44. http://dx.doi.org/10.3103/s1066369x14020066.

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6

Bassa, Alp, and Peter Beelen. "The Galois closure of Drinfeld modular towers." Journal of Number Theory 131, no. 3 (March 2011): 561–77. http://dx.doi.org/10.1016/j.jnt.2010.10.006.

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7

BIESEL, OWEN. "GALOIS CLOSURE DATA FOR EXTENSIONS OF RINGS." Transformation Groups 23, no. 1 (July 6, 2017): 41–69. http://dx.doi.org/10.1007/s00031-017-9433-x.

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8

Endo, Shizuo. "The rationality problem for norm one tori." Nagoya Mathematical Journal 202 (June 2011): 83–106. http://dx.doi.org/10.1215/00277630-1260459.

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AbstractWe consider the problem of whether the norm one torus defined by a finite separable field extension K/k is stably (or retract) rational over k. This has already been solved for the case where K/k is a Galois extension. In this paper, we solve the problem for the case where K/k is a non-Galois extension such that the Galois group of the Galois closure of K/k is nilpotent or metacyclic.

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9

Ciraulo, Francesco, and Giovanni Sambin. "A constructive Galois connection between closure and interior." Journal of Symbolic Logic 77, no. 4 (December 2012): 1308–24. http://dx.doi.org/10.2178/jsl.7704150.

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AbstractWe construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.

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10

Lehtonen, Erkko. "CHARACTERIZATION OF PRECLONES BY MATRIX COLLECTIONS." Asian-European Journal of Mathematics 03, no. 03 (September 2010): 457–73. http://dx.doi.org/10.1142/s1793557110000313.

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Preclones are described as the closed classes of the Galois connection induced by a preservation relation between operations and matrix collections. The Galois closed classes of matrix collections are also described by explicit closure conditions.

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11

Šlapal, Josef. "Galois connections between sets of paths and closure operators in simple graphs." Open Mathematics 16, no. 1 (December 31, 2018): 1573–81. http://dx.doi.org/10.1515/math-2018-0128.

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AbstractFor every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

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12

Vaughan, Theresa P. "The Normal Closure of a Quadratic Extension of a Cyclic Quartic Field." Canadian Journal of Mathematics 43, no. 5 (October 1, 1991): 1086–97. http://dx.doi.org/10.4153/cjm-1991-063-4.

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AbstractPierre Barrucand asks the following question (Unsolved Problems, # ASI 88:04, Banff, May 1988, Richard K. Guy, Ed.; also [2, p. 594]). Let K be a cyclic quartic field, and let ξ be a non-square element of K. Let M be the Galois closure of , and let G be the Galois group Gal(M/Q). Find (1) all possible G, (2) conditions on ξ to have such a G, and (3) a list of all possible subfields of M.

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13

Huang, Hau-Wen, and Wen-Ching Winnie Li. "A unified approach to the Galois closure problem." Journal of Number Theory 180 (November 2017): 251–79. http://dx.doi.org/10.1016/j.jnt.2017.04.011.

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14

Zaytsev, Alexey. "The Galois closure of the Garcia–Stichtenoth tower." Finite Fields and Their Applications 13, no. 4 (November 2007): 751–61. http://dx.doi.org/10.1016/j.ffa.2007.02.001.

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15

Vychodil, Vilem. "Closure structures parameterized by systems of isotone Galois connections." International Journal of Approximate Reasoning 91 (December 2017): 1–21. http://dx.doi.org/10.1016/j.ijar.2017.08.013.

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16

Sannai, Akiyoshi, and Anurag K. Singh. "Galois extensions, plus closure, and maps on local cohomology." Advances in Mathematics 229, no. 3 (February 2012): 1847–61. http://dx.doi.org/10.1016/j.aim.2011.12.021.

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17

Šlapel, Josef. "A GALOIS CORRESPONDENCE BETWEEN CLOSURE SPACES AND RELATIONAL SYSTEMS." Quaestiones Mathematicae 21, no. 3-4 (November 1998): 187–93. http://dx.doi.org/10.1080/16073606.1998.9632039.

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18

Yoshihara, Hisao. "Families of Galois closure curves for plane quartic curves." Journal of Mathematics of Kyoto University 43, no. 3 (2003): 651–59. http://dx.doi.org/10.1215/kjm/1250283700.

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19

Shirane, Taketo. "Families of Galois closure curves for plane quintic curves." Journal of Algebra 342, no. 1 (September 2011): 175–96. http://dx.doi.org/10.1016/j.jalgebra.2011.06.022.

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20

Gioia, Alberto. "The Galois closure for rings and some related constructions." Journal of Algebra 446 (January 2016): 450–88. http://dx.doi.org/10.1016/j.jalgebra.2015.09.027.

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21

Evdokimov, Sergei, and Ilya Ponomarenko. "Coset closure of a circulant S-ring and schurity problem." Journal of Algebra and Its Applications 15, no. 04 (February 19, 2016): 1650068. http://dx.doi.org/10.1142/s0219498816500687.

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Let [Formula: see text] be a finite group. There is a natural Galois correspondence between the permutation groups containing [Formula: see text] as a regular subgroup, and the Schur rings (S-rings) over [Formula: see text]. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group [Formula: see text] is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

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22

Engler, Antonio José. "Totally Real Rigid Elements and Galois Theory." Canadian Journal of Mathematics 50, no. 6 (December 1, 1998): 1189–208. http://dx.doi.org/10.4153/cjm-1998-058-2.

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23

CASANOVAS, E., D. LASCAR, A. PILLAY, and M. ZIEGLER. "GALOIS GROUPS OF FIRST ORDER THEORIES." Journal of Mathematical Logic 01, no. 02 (November 2001): 305–19. http://dx.doi.org/10.1142/s0219061301000119.

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We study the groups Gal L(T) and Gal KP(T), and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP (a non G-compact theory). It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

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24

Bhargava, Manjul, and Matthew Satriano. "On a notion of “Galois closure” for extensions of rings." Journal of the European Mathematical Society 16, no. 9 (2014): 1881–913. http://dx.doi.org/10.4171/jems/478.

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25

Watanabe, S. "The genera of Galois closure curves for plane quartic curves." Hiroshima Mathematical Journal 38, no. 1 (March 2008): 125–34. http://dx.doi.org/10.32917/hmj/1207580347.

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26

Garuti, Marco A. "On the \lq \lq Galois closure\rq \rq for torsors." Proceedings of the American Mathematical Society 137, no. 11 (November 1, 2009): 3575. http://dx.doi.org/10.1090/s0002-9939-09-09997-3.

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27

Zhang, Zhuo. "Constructing L-fuzzy concept lattices without fuzzy Galois closure operation." Fuzzy Sets and Systems 333 (February 2018): 71–86. http://dx.doi.org/10.1016/j.fss.2017.05.002.

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28

Müger, Michael. "Galois Theory for Braided Tensor Categories and the Modular Closure." Advances in Mathematics 150, no. 2 (March 2000): 151–201. http://dx.doi.org/10.1006/aima.1999.1860.

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29

Kozlowski, A. "Characteristic classes of Galois representations." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (November 1990): 517–22. http://dx.doi.org/10.1017/s0305004100069395.

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Let K be a field with char if K≠2 and let Ks denote the separable closure of K and GK the Galois group of the extension Ks/K. If K⊂L is a finite extension and ρ:GL↦Or(R) a (continuous) real representation of GL we have a map ρ:BGL→BO which is used to define Stiefel–Whitney classes wi(ρ) = ρ*(wi). In general if f is any element of H*(BO; ℤ/2) we denote by f(ρ) the characteristic class ρ*(f). Now letbe a genus (see e.g. [9]), for example the total Stiefel–Whitney class w = 1+w1+w2 + … Let K⊂L and ρ be as above and let denote the multiplicative transfer (see e.g. [3, 5, 2, 14, 15]). Our principal result is a generalization of theorem 1 of [3]

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30

Waterhouse, William C. "The Normal Closures of Certain Kummer Extensions." Canadian Mathematical Bulletin 37, no. 1 (March 1, 1994): 133–39. http://dx.doi.org/10.4153/cmb-1994-019-4.

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AbstractLet F be a field containing a primitive p-th root of unity, let K / F be a cyclic extension with group 〈σ〉 of order pn, and choose a in K. This paper shows how the Galois group of the normal closure of K(a1/p) over F can be determined by computations within K. The key is to define a sequence by applying the operation x ↦ σ(x)/x repeatedly to a. The first appearance of a p-th power determines the degree of the extension and restricts the Galois group to one or two possibilities. A certain expression involving that p-th root and the terms in the sequence up to that point is a p-th root of unity, and the group is finally determined by testing whether that root is 1. When (σ(a)/a G Kp, the results reduce to a theorem of A. A. Albert on cyclic extensions.

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31

Valentini, Robert C. "Quartic polynomials having Galois closure of genus zero and their value sets." Finite Fields and Their Applications 47 (September 2017): 269–75. http://dx.doi.org/10.1016/j.ffa.2017.06.013.

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32

Gajda, Wojciech, and Sebastian Petersen. "Independence of -adic Galois representations over function fields." Compositio Mathematica 149, no. 7 (April 25, 2013): 1091–107. http://dx.doi.org/10.1112/s0010437x12000711.

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AbstractLet$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

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33

Perfilieva, Irina, Ahmed A. Ramadan, and Enas H. Elkordy. "Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces." Mathematics 8, no. 8 (August 3, 2020): 1274. http://dx.doi.org/10.3390/math8081274.

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Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

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34

Singh, Anand P., and I. Perfilieva. "M-Fuzzifying Approximation Spaces, M-Fuzzifying Pretopological Spaces and M-Fuzzifying Closure Spaces." New Mathematics and Natural Computation 16, no. 03 (November 2020): 609–26. http://dx.doi.org/10.1142/s1793005720500374.

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In category theory, Galois connection plays a significant role in developing the connections among different structures. The objective of this work is to investigate the essential connections among several categories with a weaker structure than that of [Formula: see text]-fuzzifying topology, viz. category of [Formula: see text]-fuzzifying approximation spaces based on reflexive [Formula: see text]-fuzzy relations, category of [Formula: see text]-fuzzifying pretopological spaces and the category of [Formula: see text]-fuzzifying interior (closure) spaces. The interrelations among these structures are shown via the functorial diagram.

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35

MONJARDET, BERNARD. "SOME ORDER DUALITIES IN LOGIC, GAMES AND CHOICES." International Game Theory Review 09, no. 01 (March 2007): 1–12. http://dx.doi.org/10.1142/s0219198907001242.

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We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then, we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally, we present two "concrete" dualities occuring in social choice and in choice functions theories.

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36

VÂJÂITU, MARIAN, and ALEXANDRU ZAHARESCU. "AN ALGEBRAIC-METRIC EQUIVALENCE RELATION OVER p-ADIC FIELDS." Glasgow Mathematical Journal 54, no. 3 (July 31, 2012): 715–20. http://dx.doi.org/10.1017/s0017089512000468.

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AbstractLet p be a prime number, Qp the field of p-adic numbers, K a finite field extension of Qp, K a fixed algebraic closure of K and Cp the completion of K with respect to the p-adic valuation. We introduce and investigate an equivalence relation on Cp, defined in terms of field extensions and metric properties of Galois orbits over K.

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37

SHIRANE, Taketo. "GALOIS CLOSURE COVERS FOR 5-FOLD COVERS BETWEEN SMOOTH SURFACES AND ITS APPLICATION." Kyushu Journal of Mathematics 69, no. 2 (2015): 229–57. http://dx.doi.org/10.2206/kyushujm.69.229.

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38

Stalder, Nicolas. "The semisimplicity conjecture for A-motives." Compositio Mathematica 146, no. 3 (March 18, 2010): 561–98. http://dx.doi.org/10.1112/s0010437x09004448.

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AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

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39

Alexandru, Victor, Marian Vâjâitu, and Alexandru Zaharescu. "On p-adic analytic continuation with applications to generating elements." Proceedings of the Edinburgh Mathematical Society 59, no. 1 (June 10, 2015): 1–10. http://dx.doi.org/10.1017/s0013091514000376.

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AbstractGiven a prime number p and the Galois orbit O(T) of an integral transcendental element T of , the topological completion of the algebraic closure of the field of p-adic numbers, we study the p-adic analytic continuation around O(T) of functions defined by limits of sequences of restricted power series with p-adic integer coefficients. We also investigate applications to generating elements for or for some classes of closed subfields of .

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40

ZAÏMI, T., M. J. BERTIN, and A. M. ALJOUIEE. "ON NUMBER FIELDS WITHOUT A UNIT PRIMITIVE ELEMENT." Bulletin of the Australian Mathematical Society 93, no. 3 (January 11, 2016): 420–32. http://dx.doi.org/10.1017/s0004972715001410.

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We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

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41

Böckle, Gebhard, Wojciech Gajda, and Sebastian Petersen. "Independence of ℓ \ell -adic representations of geometric Galois groups." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 736 (March 1, 2018): 69–93. http://dx.doi.org/10.1515/crelle-2015-0024.

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AbstractLetkbe an algebraically closed field of arbitrary characteristic, let{K/k}be a finitely generated field extension and letXbe a separated scheme of finite type overK. For each prime{\ell}, the absolute Galois group ofKacts on the{\ell}-adic étale cohomology modules ofX. We prove that this family of representations varying over{\ell}is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure ofKof the kernels of the representations for all{\ell}become linearly disjoint over a finite extension ofK. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

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42

Van Der Ploeg, C. E. "On a Converse to the Tscgebotarev density theorem." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 44, no. 3 (June 1988): 287–93. http://dx.doi.org/10.1017/s1446788700032109.

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AbstractUsing an elementary counting procedure on biquadratic polynomials over Zp it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p → ∞. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.

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43

Kedlaya, Kiran S. "On the Geometry ofp-Typical Covers in Characteristicp." Canadian Journal of Mathematics 60, no. 1 (February 1, 2008): 140–63. http://dx.doi.org/10.4153/cjm-2008-006-8.

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AbstractForpa prime, ap-typical cover of a connected scheme on whichp= 0 is a finite étale cover whose monodromy group (i.e.,the Galois group of its normal closure) is ap-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of thep-typical quotients of the étale fundamental groups, and a decomposition theorem forp-typical covers of polynomial rings over an algebraically closed field.

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44

Gras, Georges. "On the structure of the Galois group of the Abelian closure of a number field." Journal de Théorie des Nombres de Bordeaux 26, no. 3 (2014): 635–54. http://dx.doi.org/10.5802/jtnb.883.

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45

Yakovlev, A. V. "An abstract characterization of the Galois group of the algebraic closure of a local field." Journal of Soviet Mathematics 37, no. 2 (April 1987): 1041–52. http://dx.doi.org/10.1007/bf01089101.

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46

Dyer, Matthew. "On the Weak Order of Coxeter Groups." Canadian Journal of Mathematics 71, no. 2 (January 10, 2019): 299–336. http://dx.doi.org/10.4153/cjm-2017-059-0.

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AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

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47

Orr, Martin, and Alexei N. Skorobogatov. "Finiteness theorems for K3 surfaces and abelian varieties of CM type." Compositio Mathematica 154, no. 8 (July 18, 2018): 1571–92. http://dx.doi.org/10.1112/s0010437x18007169.

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We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

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48

Breuer, Florian, and Bo-Hae Im. "Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields." Canadian Journal of Mathematics 60, no. 3 (June 1, 2008): 481–90. http://dx.doi.org/10.4153/cjm-2008-023-0.

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AbstractLetkbe a global field,a separable closure ofk, andGkthe absolute Galois groupofoverk. For everyσ ∈ Gk, letbe the fixed subfield ofunderσ. LetE/kbe an elliptic curve overk. It is known that the Mordell–Weil grouphas infinite rank. We present a new proof of this fact in the following two cases. First, when k is a global function field of odd characteristic andEis parametrized by a Drinfeld modular curve, and secondly whenkis a totally real number field andE/kis parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points onEdefined over ring class fields.

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49

Colliot-Thélène, Jean-Louis, and Alexei N. Skorobogatov. "Descente galoisienne sur le groupe de Brauer." crll 2013, no. 682 (September 6, 2012): 141–65. http://dx.doi.org/10.1515/crelle-2012-0039.

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Abstract. Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur un corps de nombres, nous donnons des estimations sur l'ordre de ce quotient. L'accouplement d'intersection entre les groupes de diviseurs et de 1-cycles modulo équivalence numérique joue ici un rôle important. For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions, e.g., over a number field, we give estimates for the order of this cokernel. We emphasise the rôle played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

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50

Zhao, Zhengjun, and Wanbao Hu. "On l-class groups of global function fields." International Journal of Number Theory 12, no. 02 (February 18, 2016): 341–56. http://dx.doi.org/10.1142/s1793042116500202.

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Abstract:

Let [Formula: see text] be a finite geometric separable extension of the rational function field [Formula: see text], and let [Formula: see text] be a finite cyclic extension of [Formula: see text] of prime degree [Formula: see text]. Assume that the ideal class number of the integral closure [Formula: see text] of [Formula: see text] in [Formula: see text] is not divisible by [Formula: see text]. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the [Formula: see text]-class group of [Formula: see text] (i.e. the Sylow [Formula: see text]-subgroup of the ideal class group of [Formula: see text]) as Galois module, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.

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