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While all quadratic fields are monogenic, already among cubic fields there are many that are not monogenic. The first example of a non-monogenic number field that was found is the
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508:{\displaystyle \mathbf {Q} (\zeta )^{+}=\mathbf {Q} (\zeta +\zeta ^{-1})}
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Analytic Theory of Algebraic Numbers
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366:{\displaystyle K=\mathbf {Q} (\zeta )}
114:Examples of monogenic fields include:
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515:is monogenic, with ring of integers
437:{\displaystyle O_{K}=\mathbf {Z} .}
278:{\displaystyle a=(1+{\sqrt {d}})/2}
228:{\displaystyle O_{K}=\mathbf {Z} }
31:for which there exists an element
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618:{\displaystyle X^{3}-X^{2}-2X-8}
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312:{\displaystyle a={\sqrt {d}}}
721:. You can help Knowledge by
555:{\displaystyle \mathbf {Z} }
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768:Algebraic number theory
717:-related article is a
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386:{\displaystyle \zeta }
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189:square-free integer
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104:minimal polynomial
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75:and the powers of
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329:Cyclotomic fields
323:≡ 2 or 3 (mod 4).
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180:{\displaystyle d}
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568:cubic field
18:mathematics
762:Categories
694:1016.11059
663:1159.11039
633:References
625:, due to
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601:−
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538:ζ
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381:ζ
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110:Examples
397:, then
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102:of the
59:. Then
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235:where
106:of α.
90:, the
24:is an
713:This
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719:stub
680:ISBN
649:ISBN
20:, a
690:Zbl
659:Zbl
336:if
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285:if
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415:=
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347:=
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331::
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300:=
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217:[
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209:=
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137:=
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121::
96:K
88:K
77:a
73:Z
65:K
61:O
57:a
53:K
49:Z
44:K
40:O
33:a
29:K
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