Unramified extensions over low degree number fields

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For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2020-07
Language
English
Article Type
Article
Citation

JOURNAL OF NUMBER THEORY, v.212, pp.72 - 87

ISSN
0022-314X
DOI
10.1016/j.jnt.2019.10.021
URI
http://hdl.handle.net/10203/273949
Appears in Collection
RIMS Journal Papers
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