### Abstract

Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p^{2} cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p^{2} extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p^{2} extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.

Original language | English |
---|---|

Pages (from-to) | 114-123 |

Number of pages | 10 |

Journal | Vestnik St. Petersburg University: Mathematics |

Volume | 51 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

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### Scopus subject areas

- Mathematics(all)

### Cite this

^{2}Extensions.

*Vestnik St. Petersburg University: Mathematics*,

*51*(2), 114-123. https://doi.org/10.3103/S1063454118020103

}

^{2}Extensions',

*Vestnik St. Petersburg University: Mathematics*, vol. 51, no. 2, pp. 114-123. https://doi.org/10.3103/S1063454118020103

**Analogue of the Hyodo Inequality for the Ramification Depth in Degree p ^{2} Extensions.** / Vostokov, S. V.; Haustov, N. V.; Zhukov, I. B.; Ivanova, O. Yu; Afanas’eva, S. S.

Research output

TY - JOUR

T1 - Analogue of the Hyodo Inequality for the Ramification Depth in Degree p 2 Extensions

AU - Vostokov, S. V.

AU - Haustov, N. V.

AU - Zhukov, I. B.

AU - Ivanova, O. Yu

AU - Afanas’eva, S. S.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p2 cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p2 extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p2 extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.

AB - Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p2 cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p2 extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p2 extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.

KW - Hyodo inequality

KW - ramification depth

UR - http://www.scopus.com/inward/record.url?scp=85048654237&partnerID=8YFLogxK

U2 - 10.3103/S1063454118020103

DO - 10.3103/S1063454118020103

M3 - Article

AN - SCOPUS:85048654237

VL - 51

SP - 114

EP - 123

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -

^{2}Extensions. Vestnik St. Petersburg University: Mathematics. 2018 Apr 1;51(2):114-123. https://doi.org/10.3103/S1063454118020103