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will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of
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are either real (and then totally real), or complex, depending on whether the
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The totally real number fields play a significant special role in
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229:, a totally imaginary quadratic extension of a totally real field
247:, London Mathematical Society Student Texts, vol. 26,
245:
Elementary theory of L-functions and
Eisenstein series
43:
send every element in the field to another element of
193:is either totally real, or contains a totally real
147:of a positive or negative number is adjoined to
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39:, and the two embeddings of the field into
170:defined by adjoining the real root will
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174:be totally real, although it is a
47:, hence the field is totally real.
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86:. Equivalent conditions are that
208:must be either totally real or
222:Totally imaginary number field
197:over which it has degree two.
1:
155:, a cubic integer polynomial
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249:Cambridge University Press
117:with the real field, over
200:Any number field that is
109:being real; or that the
280:Algebraic number theory
183:algebraic number theory
111:tensor product algebra
105:, all of the roots of
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125:to a tensor power of
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275:Field (mathematics)
100:integer polynomial
90:is generated over
49:
18:Totally real field
258:978-0-521-43569-7
210:totally imaginary
187:abelian extension
178:of real numbers.
151:. In the case of
139:of degree 2 over
35:(√2) sits inside
31:The number field
16:(Redirected from
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134:quadratic fields
82:lies inside the
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84:real numbers
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66:if for each
64:totally real
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57:number field
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160:irreducible
145:square root
269:Categories
234:References
123:isomorphic
62:is called
206:rationals
204:over the
74:into the
68:embedding
243:(1993),
227:CM-field
216:See also
195:subfield
94:by one
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202:Galois
98:of an
185:. An
176:field
162:over
121:, is
80:image
253:ISBN
96:root
78:the
55:, a
189:of
172:not
113:of
70:of
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271::
251:,
212:.
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191:Q
168:Q
164:Q
157:P
149:Q
141:Q
137:F
127:R
119:Q
115:F
107:P
103:P
92:Q
88:F
72:F
60:F
45:R
41:C
37:R
33:Q
20:)
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