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Galois Groups, Galois Representations and Iwasawa Theory
Journal of the Indian Institute of Science, 2022In this article, the author provides a lively picture of the principal questions and the main results in Iwasawa theory. After briefly explaining the role of Leopoldt's conjecture in connection with the number of independent \(\mathbb Z_p\)-extensions of number fields, she introduces Galois representations and mentions the Tate module of elliptic ...
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Canadian Journal of Mathematics, 1995
AbstractLetL/Kbe a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem overL/Kwith cyclic kernel of order 2. In this paper, we prove that if the Galois group ofL/Kis the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the ...
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AbstractLetL/Kbe a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem overL/Kwith cyclic kernel of order 2. In this paper, we prove that if the Galois group ofL/Kis the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the ...
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2023
Summary: The determination of a Galois group is an important question in computational algebraic number theory. One approach is based on the inspection of resolvents. This article reports on this method and on the performance of the current \texttt{magma} [\textit{W. Bosma}, \textit{J. Cannon} and \textit{C. Playoust}, The Magma algebra system.
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Summary: The determination of a Galois group is an important question in computational algebraic number theory. One approach is based on the inspection of resolvents. This article reports on this method and on the performance of the current \texttt{magma} [\textit{W. Bosma}, \textit{J. Cannon} and \textit{C. Playoust}, The Magma algebra system.
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The Annals of Mathematics, 1996
This paper is concerned with the connections between the Witt ring of a field and the structure of certain Galois extensions of that field. In particular, it is shown that the Witt ring determines, and is determined by, the Galois group of a certain 2-extension of the field (with an unavoidable uncertainty over the characteristic of the Witt ring in ...
Mináč, Ján, Spira, Michel
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This paper is concerned with the connections between the Witt ring of a field and the structure of certain Galois extensions of that field. In particular, it is shown that the Witt ring determines, and is determined by, the Galois group of a certain 2-extension of the field (with an unavoidable uncertainty over the characteristic of the Witt ring in ...
Mináč, Ján, Spira, Michel
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Galois Groups and Fundamental Groups
2009Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful ...
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Symplectic groups as Galois groups
Journal of Group Theory, 1998This paper proves the following result: Let \(q\) be the square of an odd prime power and let \(m \geq q\). Then \(PSp(2m,q)\) is the Galois group of a regular Galois extension of \(\mathbb{Q}(x)\). The proof is an outgrowth of the idea of rigidity, but the Nielsen classes in this instance are not rigid.
Thompson, J. G., Völklein, H.
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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2014
Two groups \(G\) and \(H\) are called isoclinic if there exist isomorphisms \(\theta: G/Z(G)\to H/Z(H)\) and \(\phi:G'\to H'\) such that \(\phi([x,y])=[x',y']\), where \(x'Z(H)=\theta(xZ(G))\) and \(y'Z(H)=\theta(yZ(G))\). (Here \(Z(G)\) is the centre of \(G\), and \(G'\) is the commutator subgroup of \(G\), generated by all commutators \([x,y]=x^{-1}y^
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Two groups \(G\) and \(H\) are called isoclinic if there exist isomorphisms \(\theta: G/Z(G)\to H/Z(H)\) and \(\phi:G'\to H'\) such that \(\phi([x,y])=[x',y']\), where \(x'Z(H)=\theta(xZ(G))\) and \(y'Z(H)=\theta(yZ(G))\). (Here \(Z(G)\) is the centre of \(G\), and \(G'\) is the commutator subgroup of \(G\), generated by all commutators \([x,y]=x^{-1}y^
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Groups, Coverings and Galois Theory
Canadian Journal of Mathematics, 1991AbstractFinite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in ...
Hansen, Vagn Lundsgaard +1 more
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1996
This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis.
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This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis.
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