# Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number theory, I do not know.

So I would appreciate suggestions of books, or chapters in a book, lecture notes, etc. that would give me an introduction. I am specifically interested in connection of cyclotomic fields and Bernoulli numbers.

Thank you.

## 3 Answers

I would just start by looking at Marcus' Number Fields for the basic algebraic number theory. It also contains tons of exercises. If you read the first 4 chapters, you should have the necessary background for most of Washington's book. I'm not familiar with Lang.

I started studying algebraic number theory last summer by going through Marcus book.

pki's suggestion is good. A couple of other books worth a look are Pollard and Diamond, The Theory of Algebraic Numbers (in the MAA Carus Mathematical Monographs series), and Stewart and Tall, Algebraic Number Theory. Ireland and Rosen, A Classical Introduction to Modern Number Theory, doesn't get as far into algebraic number theory as the others, but it is well-written and has a chapter on cyclotomic fields and a chapter on Bernoulli numbers.

• The book by I&R does not prove the unit theorem or give geometry of number methods for the finiteness of the class group (they use a different approach to bound the class number) but I think other than that it's a more comprehensive volume that Pollard and Diamond. I'd suggest looking at Samuel's Algebraic Theory of Numbers, which does have a section on the cyclotomic field generated by $p$th roots of unity for prime $p$. – KCd Nov 7 '11 at 5:29

The $n$-th Cyclotomic fields are the field $K=Q(ζ_n)$ where $ζ_n$ is a primitive $n$-th root of unity. That is, $ζ_n$ is a root of the $n$-th cyclotomic polynomial, which is the smallest polynomial factor of $x^n-1$ that is not a factor for $x^m-1$ where $m<n$. This follows from the fact that the $n$-th roots of unity are exactly the solutions to $x^n=1$ (see wiki link). If the $n$-th cyclotomic polynomial decomposes or factors into linear factors when factored over the finite field GF($p$) of order $p$, then $p=1\pmod n$. If $O_k$ is the ring of integers in $K$, then the norm of any algebraic integer $a$ in $O_k$ is a prime congruent to $1\pmod n$, or a product of prime factors congruent to $1\pmod n$.

To answer your question, I would recommend reading these course notes by Tom Lovering.

• You didn't answer the question. – Qudit Jan 20 '18 at 20:56