| Title:
|
The conductor of a cyclic quartic field using Gauss sums (English) |
| Author:
|
Spearman, Blair K. |
| Author:
|
Williams, Kenneth S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
1997 |
| Pages:
|
453-462 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that \[ K=Q\Big (\sqrt{A(D+B\sqrt{D})}\Big ), \] where \[ A \ \text{is squarefree and odd}, D=B^2+C^2 \ \text{is squarefree}, \ B>0, \ C>0, GCD(A,D) = 1. \] The conductor $f(K)$ of $K$ is $f(K) = 2^l|A|D$, where \[ l= \begin{cases} 3, \quad \text{if} \ D\equiv 2 \pmod 4 \ \text{or} \ D \equiv 1 \pmod 4, \ B \equiv 1 \pmod 2, \\ 2, \quad \text{if} \ D\equiv 1 \pmod 4, \ B \equiv 0 \pmod 2, \ A + B \equiv 3 \pmod 4, \\ 0, \quad \text{if} \ D\equiv 1 \pmod 4, \ B \equiv 0 \pmod 2, \ A + B \equiv 1 \pmod 4. \end{cases} \] A simple proof of this formula for $f(K)$ is given, which uses the basic properties of quartic Gauss sums. (English) |
| MSC:
|
11L05 |
| MSC:
|
11R16 |
| idZBL:
|
Zbl 0898.11041 |
| idMR:
|
MR1461424 |
| . |
| Date available:
|
2009-09-24T10:07:10Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127369 |
| . |
| Reference:
|
[1] K. Hardy, R.H. Hudson, D. Richman, K.S. Williams and N.M. Holtz: Calculation of the class numbers of imaginary cyclic quartic fields.Carleton-Ottawa Mathematical Lecture Note Series (Carleton University, Ottawa, Ontario, Canada), Number 7, July 1986, pp. 201. MR 0906194 |
| Reference:
|
[2] K. Ireland and M. Rosen: A Classical Introduction to Modern Number Theory.Springer-Verlag, New York, Second Edition (1990). MR 1070716 |
| . |
