USNA Pure Mathematics Seminar
The talks for the academic year 2001-2002
are held Wednesday in Chauvenet 201 at 3:45 pm unless
Antonia W. Bluher
A natural isomorphism between A6 and PSL2(9)
Let F = F3m,
where m is odd, and let a be a nonzero element of F.
Player proved through an indirect argument that the polynomial
(x6 + x)2 - a2
= (x6 + x + a)(x6 + x - a)
always has exactly one root or exactly four roots in F.
We decided to study the
polynomial x6 + x + a in order to obtain a more direct proof.
It turns out
that the polynomial is closely related to x10 + a x + 1.
are 10 ways to partition the 6 roots of x6 + x + a into
two sets of size 3,
i.e. 10 ways to factor x6 + x + a into two cubics, and
these 10 factorizations
correspond naturally to the 10 roots of x10 + a x + 1.
Using this, we can give
a direct proof of the result of Arasu and Player. Further,
by studying the
splitting fields and Galois groups of these two polynomials
(but now taking a to
be transcendental), we obtain a natural isomorphism between
group A6 (which arises as the Galois group of x6 + x + a)
and the projective
linear group PSL2(9) (which arises as the Galois group of
x10 + a x + 1).
While these two groups were known to be isomorphic, a
natural realization of the
isomorphism was not previously known.
Time: noon, Monday February 25, 2002
Reception at 11:50am in the common room on the 3rd floor of Chauvenet