# Article

 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:  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:  K. Ireland and M. Rosen: A Classical Introduction to Modern Number Theory.Springer-Verlag, New York, Second Edition (1990). MR 1070716 .

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