### Abstract

Following the analogies between knots and primes, we introduce arithmetic analogues of higher linking matrices for prime numbers by using the arithmetic Milnor numbers. As an application, we describe the Galois module structure of the l-class group of a cyclic extension of ℚ of degree l (l being a prime number) in terms of the arithmetic higher linking matrices. In particular, our formula generalizes the classical formula of Rédei on the 4 and 8 ranks of the 2-class group of a quadratic field.

Original language | English |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 669-683 |

Number of pages | 15 |

DOIs | |

Publication status | Published - Jan 1 2008 |

### Publication series

Name | Progress in Mathematics |
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Volume | 265 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Progress in Mathematics*(pp. 669-683). (Progress in Mathematics; Vol. 265). Springer Basel. https://doi.org/10.1007/978-3-7643-8608-5_16

**Milnor invariants and l-class groups.** / Morishita, Masanori.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Progress in Mathematics.*Progress in Mathematics, vol. 265, Springer Basel, pp. 669-683. https://doi.org/10.1007/978-3-7643-8608-5_16

}

TY - CHAP

T1 - Milnor invariants and l-class groups

AU - Morishita, Masanori

PY - 2008/1/1

Y1 - 2008/1/1

N2 - Following the analogies between knots and primes, we introduce arithmetic analogues of higher linking matrices for prime numbers by using the arithmetic Milnor numbers. As an application, we describe the Galois module structure of the l-class group of a cyclic extension of ℚ of degree l (l being a prime number) in terms of the arithmetic higher linking matrices. In particular, our formula generalizes the classical formula of Rédei on the 4 and 8 ranks of the 2-class group of a quadratic field.

AB - Following the analogies between knots and primes, we introduce arithmetic analogues of higher linking matrices for prime numbers by using the arithmetic Milnor numbers. As an application, we describe the Galois module structure of the l-class group of a cyclic extension of ℚ of degree l (l being a prime number) in terms of the arithmetic higher linking matrices. In particular, our formula generalizes the classical formula of Rédei on the 4 and 8 ranks of the 2-class group of a quadratic field.

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UR - http://www.scopus.com/inward/citedby.url?scp=85042855029&partnerID=8YFLogxK

U2 - 10.1007/978-3-7643-8608-5_16

DO - 10.1007/978-3-7643-8608-5_16

M3 - Chapter

AN - SCOPUS:85042855029

T3 - Progress in Mathematics

SP - 669

EP - 683

BT - Progress in Mathematics

PB - Springer Basel

ER -