Essential Algebraic Number Theory – Ivan Fesenko

Putting Y = X−1 and performing the calculations in the field L((Y)), we consequently deduce f(X) = Y −p(1+ap−1Y +···+a0Y p), f(X) = Y p 1+ap−1Y +···+a0Y p ≡Y p mod Y p+1, X −σi(πL) = 1−σi(πL)Y = ∑ j⩾0 σi(π j L)Y j+1 (because 1/(1−Y) = ∑i⩾0Y i in F((Y))). Hence ∑ j⩾0 p−1 ∑ i=0 σi(π j L)Y j+1 f ′σi(πL)
≡Y p mod Y p+1, or TrL/F π j f ′(πL) ! = p−1 ∑ i=0 σi(π j f ′σi(πL)
= ( if 0 ⩽j ⩽p−2, if j = p−1, as desired.

□ PROPOSITION. Let [a] denote the maximal integer ⩽a. For an integer i ⩾0 put j(i) = s+1+[(i−1−s)/p]. Then TrL/F(πi LOL) = π j(i) Proof. One has f ′(πL) = ∏p−1 i=1 πL −σi(πL)
. From the definition of s we deduce σi(πL)/πL ≡1+iηπs mod πs+1 . Then f ′(πL) = (p−1)!(−η)p−1π(p−1)(s+1) ε with some ε ∈1+M (p−1)(s+1)+1 . Since F = L, for a prime element πF in F one has the representation πF = π p Lε′ with ε′ ∈UL.

The previous Lemma implies TrL/F  π j+s+1 εj+s+1  = ( if 0 ⩽j < p−1, πs+1 if j = p−1 for εj+s+1 = (ε′)s+1/ (p−1)!(−η)p−1ε
. Since TrL/F(πi Fα) = πi F TrL/F(α), we can choose the units εj+s+1, for every integer j > 0, such that TrL/F(π j+s+1 εj+s+1) = 0 if p∤(j +1) and = πs+(j+1)/p if p|(j +1).

Thus, since the OF-module πi LOL is generated by π j Lεj, j ⩾i, we conclude that TrL/F(πi LOL) = π j(i) □ 13.5. PROPOSITION. Let L/F be a totally ramified Galois extension of de- gree p = char(F) > 0. Let πL be a prime element in L. Then πF = NL/FπL is a prime element in F.

Series Editors: Shigeru Kanemitsu (Shandong University, PR China & Kerala School of Mathematics, India) Jianya Liu (Shandong University, PR China) Editorial Board Members: R. Balasubramanian (Institute of Mathematical Sciences, India) V. N. Chubarikov (Moscow State University, Russian Federation) Christopher Deninger (Universität Münster, Germany) Chaohua Jia (Chinese Academy of Sciences, PR China) H. Niederreiter (RICAM, Austria) Advisory Board: Current Member M. Waldschmidt (Université Pierre et Marie Curie, France) In Memoriam K. Ramachandra (Tata Institute of Fundamental Research, Mumbai-Bangalore and National Institute of Advanced Studies, Bangalore, India) A. Schinzel (Polish Academy of Sciences, Poland) *For the complete list of the published titles in this series, please visit: www.worldscientific.com/series/sntia World Scientific Published by World Scientific Publishing Co.

Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Control Number: 2025054220 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover image: Delicate Tension, No. 85 (1923) by Wassily Kandinsky. Public domain. Series on Number Theory and Its Applications — Vol. 18 ESSENTIAL ALGEBRAIC NUMBER THEORY Copyright © 2026 by Ivan Fesenko All rights reserved. ISBN 978-981-98-2571-4 (hardcover) ISBN 978-981-98-2681-0 (paperback) ISBN 978-981-98-2572-1 (ebook for institutions) ISBN 978-981-98-2573-8 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/14658#t=suppl Desk Editors: Nambirajan Karuppiah/Lai Fun Kwong Typeset by Stallion Press Email: [email protected] Printed in Singapore Preface This book is based on courses given in Russia, the UK and China, as well as numerous talks delivered in Germany, Japan, France, and the USA.

The most recent lecture courses were given at Tsinghua University in 2023–2024 and at Westlake University in 2025. The material of the first chapter was lectured for over 10 years in the UK. The book offers a fast and relatively easy introduction into the main aspects of algebraic number theory, including its basic aspects, main results about local and global fields and of explicit class field theory. The emphasise is on a clear presentation, using as little auxiliary tools from algebra or analysis as possible.

The presentation of this book was motivated by the author’s study and research in generalisations of class field theory such as higher class field theory, anabelian geometry and some topics in the Langlands program. The main aim of the book is the enable the reader to learn central key aspects of the theory and to prepare for research work in modern number theory.

The first chapter is a compact simplified presentation of core features of alge- braic number fields, without using analytic tools. Various properties of algebraic integers are deduced from properties of Dedekind rings.

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.

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