In mathematics , the conductor-discriminant formula or Führerdiskriminantenproduktformel , introduced by Hasse (1926 , 1930 ) for abelian extensions and by Artin (1931 ) for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension
L
/
K
{\displaystyle L/K}
of local or global fields from the Artin conductors of the irreducible characters
I
r
r
(
G
)
{\displaystyle \mathrm {Irr} (G)}
of the Galois group
G
=
G
(
L
/
K
)
{\displaystyle G=G(L/K)}
.
Statement
Let
L
/
K
{\displaystyle L/K}
be a finite Galois extension of global fields with Galois group
G
{\displaystyle G}
. Then the discriminant equals
d
L
/
K
=
∏ ∏ -->
χ χ -->
∈ ∈ -->
I
r
r
(
G
)
f
(
χ χ -->
)
χ χ -->
(
1
)
,
{\displaystyle {\mathfrak {d}}_{L/K}=\prod _{\chi \in \mathrm {Irr} (G)}{\mathfrak {f}}(\chi )^{\chi (1)},}
where
f
(
χ χ -->
)
{\displaystyle {\mathfrak {f}}(\chi )}
equals the global Artin conductor of
χ χ -->
{\displaystyle \chi }
.
Example
Let
L
=
Q
(
ζ ζ -->
p
n
)
/
Q
{\displaystyle L=\mathbf {Q} (\zeta _{p^{n}})/\mathbf {Q} }
be a cyclotomic extension of the rationals. The Galois group
G
{\displaystyle G}
equals
(
Z
/
p
n
)
× × -->
{\displaystyle (\mathbf {Z} /p^{n})^{\times }}
. Because
(
p
)
{\displaystyle (p)}
is the only finite prime ramified, the global Artin conductor
f
(
χ χ -->
)
{\displaystyle {\mathfrak {f}}(\chi )}
equals the local one
f
(
p
)
(
χ χ -->
)
{\displaystyle {\mathfrak {f}}_{(p)}(\chi )}
. Because
G
{\displaystyle G}
is abelian, every non-trivial irreducible character
χ χ -->
{\displaystyle \chi }
is of degree
1
=
χ χ -->
(
1
)
{\displaystyle 1=\chi (1)}
. Then, the local Artin conductor of
χ χ -->
{\displaystyle \chi }
equals the conductor of the
p
{\displaystyle {\mathfrak {p}}}
-adic completion of
L
χ χ -->
=
L
k
e
r
(
χ χ -->
)
/
Q
{\displaystyle L^{\chi }=L^{\mathrm {ker} (\chi )}/\mathbf {Q} }
, i.e.
(
p
)
n
p
{\displaystyle (p)^{n_{p}}}
, where
n
p
{\displaystyle n_{p}}
is the smallest natural number such that
U
Q
p
(
n
p
)
⊆ ⊆ -->
N
L
p
χ χ -->
/
Q
p
(
U
L
p
χ χ -->
)
{\displaystyle U_{\mathbf {Q} _{p}}^{(n_{p})}\subseteq N_{L_{\mathfrak {p}}^{\chi }/\mathbf {Q} _{p}}(U_{L_{\mathfrak {p}}^{\chi }})}
. If
p
>
2
{\displaystyle p>2}
, the Galois group
G
(
L
p
/
Q
p
)
=
G
(
L
/
Q
p
)
=
(
Z
/
p
n
)
× × -->
{\displaystyle G(L_{\mathfrak {p}}/\mathbf {Q} _{p})=G(L/\mathbf {Q} _{p})=(\mathbf {Z} /p^{n})^{\times }}
is cyclic of order
φ φ -->
(
p
n
)
{\displaystyle \varphi (p^{n})}
, and by local class field theory and using that
U
Q
p
/
U
Q
p
(
k
)
=
(
Z
/
p
k
)
× × -->
{\displaystyle U_{\mathbf {Q} _{p}}/U_{\mathbf {Q} _{p}}^{(k)}=(\mathbf {Z} /p^{k})^{\times }}
one sees easily that if
χ χ -->
{\displaystyle \chi }
factors through a primitive character of
(
Z
/
p
i
)
× × -->
{\displaystyle (\mathbf {Z} /p^{i})^{\times }}
, then
f
(
p
)
(
χ χ -->
)
=
p
i
{\displaystyle {\mathfrak {f}}_{(p)}(\chi )=p^{i}}
whence as there are
φ φ -->
(
p
i
)
− − -->
φ φ -->
(
p
i
− − -->
1
)
{\displaystyle \varphi (p^{i})-\varphi (p^{i-1})}
primitive characters of
(
Z
/
p
i
)
× × -->
{\displaystyle (\mathbf {Z} /p^{i})^{\times }}
we obtain from the formula
d
L
/
Q
=
(
p
φ φ -->
(
p
n
)
(
n
− − -->
1
/
(
p
− − -->
1
)
)
)
{\displaystyle {\mathfrak {d}}_{L/\mathbf {Q} }=(p^{\varphi (p^{n})(n-1/(p-1))})}
, the exponent is
∑ ∑ -->
i
=
0
n
(
φ φ -->
(
p
i
)
− − -->
φ φ -->
(
p
i
− − -->
1
)
)
i
=
n
φ φ -->
(
p
n
)
− − -->
1
− − -->
(
p
− − -->
1
)
∑ ∑ -->
i
=
0
n
− − -->
2
p
i
=
n
φ φ -->
(
p
n
)
− − -->
p
n
− − -->
1
.
{\displaystyle \sum _{i=0}^{n}(\varphi (p^{i})-\varphi (p^{i-1}))i=n\varphi (p^{n})-1-(p-1)\sum _{i=0}^{n-2}p^{i}=n\varphi (p^{n})-p^{n-1}.}
Notes
References
Artin, Emil (1931), "Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper." , Journal für die Reine und Angewandte Mathematik (in German), 1931 (164): 1–11, doi :10.1515/crll.1931.164.1 , ISSN 0075-4102 , S2CID 117731518 , Zbl 0001.00801
Hasse, H. (1926), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie." , Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 35 : 1–55
Hasse, H. (1930), "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper." , Journal für die reine und angewandte Mathematik (in German), 1930 (162): 169–184, doi :10.1515/crll.1930.162.169 , ISSN 0075-4102 , S2CID 199546442
Neukirch, Jürgen (1999). Algebraische Zahlentheorie . Grundlehren der mathematischen Wissenschaften . Vol. 322. Berlin: Springer-Verlag . ISBN 978-3-540-65399-8 . MR 1697859 . Zbl 0956.11021 .