WebKummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. Webis what motivated Ernst Kummer to develop his theory of ideal numbers, which restores unique factorization for the rings in question. To begin a study of this theory, we start by …
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WebKummer worked out the arithmetic of cyclotomic extensions guided by his desire to find the higher reciprocity laws; notions such as unique factorization into ideal numbers, the ideal class group, units, the Stickelberger relation, Hilbert 90, norm residues and Kummer extensions owe their existence to his work on reciprocity laws. In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The … See more A Kummer extension is a field extension L/K, where for some given integer n > 1 we have • K contains n distinct nth roots of unity (i.e., roots of X − 1) • L/K has abelian Galois group of See more Suppose that G is a profinite group acting on a module A with a surjective homomorphism π from the G-module A to itself. Suppose also that G acts trivially on the kernel C of π … See more One of the main tools in Kummer theory is the Kummer map. Let $${\displaystyle m}$$ be a positive integer and let $${\displaystyle K}$$ be a field, not necessarily containing the $${\displaystyle m}$$th roots of unity. Letting See more • Quadratic field See more b. sporothermodurans
O arXiv:1908.01152v4 [math.NT] 21 Oct 2024 - ResearchGate
WebApr 3, 2013 · Kummer's Theorem for cyclotomic units. Let $A=\Bbb {Z} [\zeta_n]$ be the ring of integers of the $n$ - th cyclotomic cyclotomic field for $n=32$. It is true that the unit … Webtheorists’ interest for a long time. Among them, Kummer accomplished a monu-mental work on ideal class groups of cyclotomic fields in the 19th century toward Fermat’s Last Theorem. Kummer studied the ideal class group Cl(Q(µp)) of the p-th cyclotomic field Q(µp), where p is an odd prime number and µp the group of p-th roots of unity. WebAug 3, 2024 · The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of magnitude (a simple … bsp online internet personal banking