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are examples. A cyclotomic extension, under either definition, is always abelian.
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167:-th roots and on the roots of unity, giving a non-abelian Galois group as
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gives a complete description of the abelian extension case, and the
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provides detailed information about the abelian extensions of
58:. Going in the other direction, a Galois extension is called
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There are two slightly different definitions of the term
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to a field, or a subextension of such an extension. The
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It can mean either an extension formed by adjoining
70:of an abelian group. Every finite extension of a
27:Galois extension whose Galois group is abelian
159:). In general, however, the Galois groups of
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163:-th roots of elements operate both on the
190:There is an important analogy with the
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202:which relates directly to the first
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54:, the extension is also called a
50:. When the Galois group is also
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139:is an abelian extension (if
229:Encyclopedia of Mathematics
135:is adjoined, the resulting
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131:-th root of an element of
127:-th root of unity and the
97:over finite fields, and
274:Algebraic number theory
222:Kuz'min, L.V. (2001) ,
177:Kronecker–Weber theorem
74:is a cyclic extension.
62:if its Galois group is
224:"cyclotomic extension"
209:Further information:
123:contains a primitive
106:cyclotomic extension.
18:Cyclotomic extension
245:"Abelian Extension"
169:semi-direct product
157:separable extension
147:we should say that
143:has characteristic
279:Class field theory
242:Weisstein, Eric W.
83:Class field theory
192:fundamental group
179:tells us that if
114:cyclotomic fields
36:abelian extension
16:(Redirected from
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269:Field extensions
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211:Ring class field
185:rational numbers
183:is the field of
137:Kummer extension
95:algebraic curves
56:cyclic extension
40:Galois extension
32:abstract algebra
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99:local fields
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72:finite field
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44:Galois group
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119:If a field
78:Description
263:Categories
216:References
68:extensions
250:MathWorld
234:EMS Press
196:topology
64:solvable
60:solvable
48:abelian
171:. The
52:cyclic
42:whose
38:is a
34:, an
194:in
93:of
46:is
30:In
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181:K
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149:p
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141:K
133:K
129:n
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121:K
20:)
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