### Abstract

Let F, K and L be algebraic number fields such that {Mathematical expression}, [K:F]=2 and [L:K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields {Mathematical expression} with [K:F]=2, [L:K]=2 where L is unramified over K, but L is not normal over F.

Original language | English (US) |
---|---|

Pages (from-to) | 343-349 |

Number of pages | 7 |

Journal | Manuscripta Mathematica |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1979 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**A note on the normality of unramified, abelian extensions of quadratic extensions.** / Madden, Daniel; Velez, William Yslas.

Research output: Contribution to journal › Article

*Manuscripta Mathematica*, vol. 30, no. 4, pp. 343-349. https://doi.org/10.1007/BF01301254

}

TY - JOUR

T1 - A note on the normality of unramified, abelian extensions of quadratic extensions

AU - Madden, Daniel

AU - Velez, William Yslas

PY - 1979/12

Y1 - 1979/12

N2 - Let F, K and L be algebraic number fields such that {Mathematical expression}, [K:F]=2 and [L:K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields {Mathematical expression} with [K:F]=2, [L:K]=2 where L is unramified over K, but L is not normal over F.

AB - Let F, K and L be algebraic number fields such that {Mathematical expression}, [K:F]=2 and [L:K]=n. It is a simple consequence of the class field theory that, if L is an abelian, unramified extension of K and (n,h)=1, where h is the class number of F, then L is normal over F. The purpose of this note is to point out the necessity of the condition (n,h)=1 by constructing for any field F with even class number a tower of fields {Mathematical expression} with [K:F]=2, [L:K]=2 where L is unramified over K, but L is not normal over F.

UR - http://www.scopus.com/inward/record.url?scp=0041951492&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041951492&partnerID=8YFLogxK

U2 - 10.1007/BF01301254

DO - 10.1007/BF01301254

M3 - Article

AN - SCOPUS:0041951492

VL - 30

SP - 343

EP - 349

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 4

ER -