On the minimal polynomial of Gauss periods for prime powers
Authors: Gurak S. 
Published: August 2006

For a positive integer

m
m

, set

ζ

m

=
exp
⁡
(
2
π
i

/

m
)

\zeta _{m}=\exp (2\pi i/m)

and let

Z

m

∗

\textbf {Z}_{m}^{*}

denote the group of reduced residues modulo

m
m

. Fix a congruence group

H
H

of conductor

m
m

and of order

f
f

. Choose integers

t

1

,
…
,

t

e

t_{1},\dots ,t_{e}

to represent the

e
=
ϕ
(
m
)

/

f

e=\phi (m)/f

cosets of

H
H

in

Z

m

∗

\textbf {Z}_{m}^{*}

. The Gauss periods
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row theta Subscript j Baseline equals sigma-summation Underscript x element-of upper H Endscripts zeta Subscript m Superscript t Super Subscript j Superscript x Baseline left-parenthesis 1 less-than-or-equal-to j less-than-or-equal-to e right-parenthesis EndLayout">

θ

j

=

∑

x
∈
H

ζ

m

t

j

x

(
1
≤
j
≤
e
)

\displaylines { \theta _{j} =\sum _{x \in H} \zeta _{m}^{t_{j}x} \;\;\; (1 \leq j \leq e) }

\]
corresponding to

H
H

are conjugate and distinct over

Q

\textbf {Q}

with minimal polynomial
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row g left-parenthesis x right-parenthesis equals x Superscript e Baseline plus c 1 x Superscript e minus 1 Baseline plus midline-horizontal-ellipsis plus c Subscript e minus 1 Baseline x plus c Subscript e Baseline period EndLayout">

g
(
x
)
=

x

e

+

c

1

x

e
−
1

+
⋯
+

c

e
−
1

x
+

c

e

.

\displaylines { g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. }

\]
To determine the coefficients of the period polynomial

g
(
x
)

g(x)

(or equivalently, its reciprocal polynomial

G
(
X
)
=

X

e

g
(

X

−
1

)
)

G(X)=X^{e}g(X^{-1}))

is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case

m
=
p

m=p

, an odd prime, with

f
>
1

f >1

fixed. In this setting, it is known for certain integral power series

A
(
X
)

A(X)

and

B
(
X
)

B(X)

, that for any positive integer

N
N

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper G left-parenthesis upper X right-parenthesis identical-to upper A left-parenthesis upper X right-parenthesis dot upper B left-parenthesis upper X right-parenthesis Superscript StartFraction p minus 1 Over f EndFraction Baseline left-parenthesis mod upper X Superscript upper N Baseline right-parenthesis EndLayout">

G
(
X
)
≡
A
(
X
)
⋅
B
(
X

)

p
−
1

f

(

mod

X

N

)

\displaylines { G(X) \equiv A(X)\cdot B(X)^{\frac {p-1}{f}} \;\;\;(\textrm {mod}\;X^{N}) }

\]
holds in

Z

[
X
]

\textbf {Z}[X]

for all primes

p
≡
1
(

mod

f
)

p \equiv 1(\textrm {mod}\; f)

except those in an effectively determinable finite set. Here we describe an analogous result for the case

m
=

p

α

m=p^{\alpha }

, a prime power (

α
>
1

\alpha > 1

). The methods extend for odd prime powers

p

α

p^{\alpha }

to give a similar result for certain twisted Gauss periods of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row psi Subscript j Baseline equals i Superscript asterisk Baseline StartRoot p EndRoot sigma-summation Underscript x element-of upper H Endscripts left-parenthesis StartFraction t Subscript j Baseline x Over p EndFraction right-parenthesis zeta Subscript p Sub Superscript alpha Subscript Superscript t Super Subscript j Superscript x Baseline left-parenthesis 1 less-than-or-equal-to j less-than-or-equal-to e right-parenthesis comma EndLayout">

ψ

j

=

i

∗

p

∑

x
∈
H

(

t

j

x

p

)

ζ

p

α

t

j

x

(
1
≤
j
≤
e
)
,

\displaylines { \psi _{j} = i^{*} \sqrt {p} \sum _{x \in H} (\frac {t_{j}x}{p}) \zeta _{p^{\alpha }}^{t_{j}x} \;\;(1 \leq j \leq e),}

\]
where

(

p

)

(\frac { }{p})

denotes the usual Legendre symbol and

i

∗

=

i

(
p
−
1

)

2

4

i^{*}= i^{\frac {(p-1)^{2}}{4}}

.