An Embedding Problem with Cyclic Kernel |
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Authors: | V. V. Ishkhanov B. B. Lur’e |
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Affiliation: | (1) St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia |
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Abstract: | The problem of embedding a quadratic extension of a number field into an extension with a cyclic 2-group is studied. We prove a reduction theorem showing that, under the compatibility condition, an additional embedding condition consists of the solvability of a problem with cyclic kernel of order 16 (of course, if the degree of the required field is no less than 16). An additional condition of embedding into a field of degree 16 is found; namely, the number generating the given quadratic extension must be a norm in a cyclotomic field containing the primitive eighth roots of unity. For Q, the embedding condition is simpler: all the odd prime divisors of the generating element must be congruent with 1 modulo the order of the extension group. In addition, the quadratic extension must be real. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 144–152. |
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