The shortest vector problem and tame kernels of cyclotomic fields. (English) Zbl 1470.11320
Summary: The Shortest Vector Problem and the computation of the tame kernel in algebraic K-theory are two different but very important problems, and up to now no one has found any manipulated relations between them. In this paper, we are successful for the first time to reduce the computation of tame kernel, in particular the computation of elements of the set \(C_m\), the key part in the method proposed by Tate, to the Shortest Vector Problem of some lattice. Then, with the help of PARI library we implement the above idea and develop a program for computing the tame kernels of the cyclotomic fields with class number 1, and as an example, by running the program we obtain the tame kernel of the cyclotomic field \(\mathbb{Q}(\zeta_7)\).
MSC:
| 11Y40 | Algebraic number theory computations |
| 11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |
| 11R18 | Cyclotomic extensions |
| 11R70 | \(K\)-theory of global fields |
| 19F15 | Symbols and arithmetic (\(K\)-theoretic aspects) |
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