The conductor of a cyclic quartic field using Gauss sums

Spearman, Blair K., and Williams, Kenneth S.. "The conductor of a cyclic quartic field using Gauss sums." Czechoslovak Mathematical Journal 47.3 (1997): 453-462. <http://eudml.org/doc/30375>.

@article{Spearman1997,
abstract = {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 \[ l= \{\left\lbrace \begin\{array\}\{ll\} 3, \quad \text\{if\} \ D\equiv 2 \hspace\{10.0pt\}(\@mod \; 4) \ \text\{or\} \ D \equiv 1 \hspace\{10.0pt\}(\@mod \; 4), \ B \equiv 1 \hspace\{10.0pt\}(\@mod \; 2), \\ 2, \quad \text\{if\} \ D\equiv 1 \hspace\{10.0pt\}(\@mod \; 4), \ B \equiv 0 \hspace\{10.0pt\}(\@mod \; 2), \ A + B \equiv 3 \hspace\{10.0pt\}(\@mod \; 4), \\ 0, \quad \text\{if\} \ D\equiv 1 \hspace\{10.0pt\}(\@mod \; 4), \ B \equiv 0 \hspace\{10.0pt\}(\@mod \; 2), \ A + B \equiv 1 \hspace\{10.0pt\}(\@mod \; 4). \end\{array\}\right.\} \] A simple proof of this formula for $f(K)$ is given, which uses the basic properties of quartic Gauss sums.},
author = {Spearman, Blair K., Williams, Kenneth S.},
journal = {Czechoslovak Mathematical Journal},
keywords = {cyclic quartic extension; conductor; Gauss sums},
language = {eng},
number = {3},
pages = {453-462},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The conductor of a cyclic quartic field using Gauss sums},
url = {http://eudml.org/doc/30375},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Spearman, Blair K.
AU - Williams, Kenneth S.
TI - The conductor of a cyclic quartic field using Gauss sums
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 3
SP - 453
EP - 462
AB - 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 \[ l= {\left\lbrace \begin{array}{ll} 3, \quad \text{if} \ D\equiv 2 \hspace{10.0pt}(\@mod \; 4) \ \text{or} \ D \equiv 1 \hspace{10.0pt}(\@mod \; 4), \ B \equiv 1 \hspace{10.0pt}(\@mod \; 2), \\ 2, \quad \text{if} \ D\equiv 1 \hspace{10.0pt}(\@mod \; 4), \ B \equiv 0 \hspace{10.0pt}(\@mod \; 2), \ A + B \equiv 3 \hspace{10.0pt}(\@mod \; 4), \\ 0, \quad \text{if} \ D\equiv 1 \hspace{10.0pt}(\@mod \; 4), \ B \equiv 0 \hspace{10.0pt}(\@mod \; 2), \ A + B \equiv 1 \hspace{10.0pt}(\@mod \; 4). \end{array}\right.} \] A simple proof of this formula for $f(K)$ is given, which uses the basic properties of quartic Gauss sums.
LA - eng
KW - cyclic quartic extension; conductor; Gauss sums
UR - http://eudml.org/doc/30375
ER -