Affiliation:
1. Institute of Mathematics, Vilnius University, Naugarduko 24 , Vilnius LT-03225 , Lithuania
Abstract
Abstract
In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over
Q
{\mathbb{Q}}
cannot be of degree 8. This completes the classification of so-called product-feasible triplets
(
a
,
b
,
c
)
∈
N
3
\left(a,b,c)\in {{\mathbb{N}}}^{3}
with
a
≤
b
≤
c
a\le b\le c
and
b
≤
7
b\le 7
. The triplet
(
a
,
b
,
c
)
\left(a,b,c)
is called product-feasible if there are algebraic numbers
α
,
β
\alpha ,\beta
, and
γ
\gamma
of degrees
a
,
b
a,b
, and
c
c
over
Q
{\mathbb{Q}}
, respectively, such that
α
β
=
γ
\alpha \beta =\gamma
. In the proof, we use a proposition that describes all monic quartic irreducible polynomials in
Q
[
x
]
{\mathbb{Q}}\left[x]
with four roots of equal moduli and is of independent interest. We also prove a more general statement, which asserts that for any integers
n
≥
2
n\ge 2
and
k
≥
1
k\ge 1
, the triplet
(
a
,
b
,
c
)
=
(
n
,
(
n
−
1
)
k
,
n
k
)
\left(a,b,c)=\left(n,\left(n-1)k,nk)
is product-feasible if and only if
n
n
is a prime number. The choice
(
n
,
k
)
=
(
4
,
2
)
\left(n,k)=\left(4,2)
recovers the case
(
a
,
b
,
c
)
=
(
4
,
6
,
8
)
\left(a,b,c)=\left(4,6,8)
as well.