Swinnerton-Dyer polynomial
family of polynomials
In algebra, the Swinnerton-Dyer polynomials are a family of polynomials, introduced by Peter Swinnerton-Dyer, that serve as examples where polynomial factorization algorithms have worst-case runtime. They have the property of being reducible modulo every prime, while being irreducible over the rational numbers. They are a standard counterexample in number theory.
Given a finite set
P
{\displaystyle P}
of prime numbers, the Swinnerton-Dyer polynomial associated to
P
{\displaystyle P}
is the polynomial:
f
P
(
x
)
=
∏
(
x
+
∑
p
∈
P
(
±
)
p
)
{\displaystyle f_{P}(x)=\prod \left(x+\sum _{p\in P}(\pm ){\sqrt {p}}\right)}
where the product extends over all
2
|
P
|
{\displaystyle 2^{|P|}}
choices of sign in the enclosed sum. The polynomial
f
P
(
x
)
{\displaystyle f_{P}(x)}
has degree
2
|
P
|
{\displaystyle 2^{|P|}}
and integer coefficients, which alternate in sign. If
|
P
|
>
1
{\displaystyle |P|>1}
, then
f
P
(
x
)
{\displaystyle f_{P}(x)}
is reducible modulo
p
{\displaystyle p}
for all primes
p
{\displaystyle p}
, into linear and quadratic factors, but irreducible over
Q
{\displaystyle \mathbb {Q} }
. The Galois group of
f
P
(
x
)
{\displaystyle f_{P}(x)}
is
Z
2
|
P
|
{\displaystyle \mathbb {Z} _{2}^{|P|}}
.
The first few Swinnerton-Dyer polynomials are:
f
{
2
}
(
x
)
=
x
2
−
2
=
(
x
−
2
)
(
x
+
2
)
{\displaystyle f_{\{2\}}(x)=x^{2}-2=(x-{\sqrt {2}})(x+{\sqrt {2}})}
f
{
2
,
3
}
(
x
)
=
x
4
−
10
x
2
+
1
=
(
x
−
2
−
3
)
(
x
−
2
+
3
)
(
x
+
2
−
3
)
(
x
+
2
+
3
)
{\displaystyle f_{\{2,3\}}(x)=x^{4}-10x^{2}+1=(x-{\sqrt {2}}-{\sqrt {3}})(x-{\sqrt {2}}+{\sqrt {3}})(x+{\sqrt {2}}-{\sqrt {3}})(x+{\sqrt {2}}+{\sqrt {3}})}
f
{
2
,
3
,
5
}
(
x
)
=
x
8
−
40
x
6
+
352
x
4
−
960
x
2
+
576.
{\displaystyle f_{\{2,3,5\}}(x)=x^{8}-40x^{6}+352x^{4}-960x^{2}+576.}
References
von zur Gathen, Joachim; Gerhard, Jürgen (April 2013). Modern Computer Algebra (Third ed.).
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