COURSE OVERVIEW
One of the major goals of this course is to understand the structure of roots of
high degree polynomial equations. We are all familiar with the quadratic formula
?
´b ˘ b2 ´ 4ac
2a
for the roots of the quadratic polynomial equation ppxq :“ ax2 ` bx ` c “ 0. This
simple formula is a useful tool allowing one to express the roots of p as a radical
expression of the coefficients; that is, a composition of rational functions and roots
in a, b, c.
A similar formula is available for the roots of cubic equations ppxq :“ ax3 `
2
bx ` cx ` d “ 0, although it is not nearly so succinct. In particular, the three roots
x1 , x2 , x3 P C are given by
ˆ
˙
b2 ´ 3ac
1
b `A `
,
x1 “ ´
3a
A
?
?
ˆ
˙
` ´1 ` i 3 ˘´1 b2 ´ 3ac
1
´1 ` i 3
x2 “ ´
A `
b `
,
3a
2
2
A
?
?
ˆ
˙
` ´1 ´ i 3 ˘´1 b2 ´ 3ac
1
´1 ´ i 3
x3 “ ´
b `
A `
,
3a
2
2
A
where
d
a
´27 a2 p18abcd ´ 4b3 d ` b2 c2 ´ 4ac3 ´ 27a2 d2 q
A“
.
2
Don’t worry, you won’t be expected to remember this expression! Rather than the
ability to compute the xj , we’re interested in the theoretical observation that such
an formula exists, again involving only combinations of rational functions and roots
in a, b, c, d.
Finally, we consider quartic equations ppxq :“ ax4 `bx3 `cx2 `dx`e “ 0. Again,
one can find a general formula which provides a radical expression for the roots of
p in terms of the coefficients. As you’d expect, this formula is rather unwieldy and
we will refrain from reproducing it here.
What about higher order polynomials? Can we find a similar formula for the
roots of quintic polynomials ppxq :“ ax5 ` bx4 ` cx3 ` dx2 ` ex ` f ?
It transpires that no general formula is available for polynomials of degree 5 (the
Abel-Ruffini theorem). In fact, a stronger result is known: there exist polynomials
of degree 5 for which it is impossible to express the roots in terms of radical expressions! (Do you understand the distinction between the two results?) The latter
result is due to Galois, who introduced the algebraic framework used to study these
problems which later developed into what is now known as Galois theory.
Galois theory involves a beautiful interplay between field and group theory. In
the context of studying polynomial equations, the groups arise from symmetries
associated to the polynomial in question. As a rough indication of the connection,
consider a general n degree monic polynomial equation (so here an “ 1)
n
ÿ
ppxq “
aj xj .
3
2b3 ´ 9abc ` 27a2 d `
j“0
1
2
COURSE OVERVIEW
We know by the fundamental theorem of algebra that there are n roots ξ1 , . . . , ξn
(counted with multiplicity) and we can write p as
n
ź
ppxq “
px ´ ξj q.
j“1
From this we see that the coefficients a0 , a1 , . . . , an are given by
aj “ p´1qn´j en´j pξ1 , . . . , ξn q
where ej is the elementary symmetric polynomials of degree j in n-variables. Explicitly,
ÿ
ej pX1 , . . . , Xn q :“
Xi1 . . . Xij .
i1 㨨¨ăij
These polynomials have the nice property that they are invariant under permutations of the variables. In particular, if we let σ P Sn , then
ej pXσp1q , . . . , Xσpnq q “ ej pX1 , . . . , Xn q.
In fact, the ej are very special in the sense that every polynomials with this invariance property can be expressed in terms of the ej .
Definition. Let R be a ring. We say P P RrX1 , . . . , Xn s is a symmetric polynomial
if
P pXσp1q , . . . , Xσpnq q “ P pX1 , . . . , Xn q
for all σ P Sn .
It turns out that every symmetric polynomial can be written as a polynomial in
the ej .
Theorem (Fundamental theorem of symmetric polynomials). Suppose P P RrX1 , . . . , Xn s
is symmetric. Then there exists some Q P RrX1 , . . . , Xn s such that
P pX1 , . . . , Xn q “ Qpe1 pX1 , . . . , Xn q, . . . , en pX1 , . . . , Xn qq.
This suggests that if we want to determine an expression for the roots of p
given the coefficients, then the theory of the symmetric groups Sn could play a
rôle. Indeed, the fact that general formulae exist for the roots of polynomials of
degrees 1 to 4 but there is no formula for quintic equations can be understood by
the comparing the structure of the groups Sj for j “ 1, 2, 3, 4 with the structure
of S5 . We’ll see that S5 has a relatively complicated structure which prohibits the
existence of such a formula (the key idea here is that of a solvable group).
We’ll see some rudiments of these ideas by deducing the quadratic formula. One
can easy reduce to the case where a “ 1 and so we wish to find the roots of
x2 ` bx ` c. Let ξ1 , ξ2 denote these roots. We know
b “ ´e1 pξ1 , ξ2 q “ ´pξ1 ` ξ2 q and c “ e2 pξ1 , ξ2 q “ ξ1 ξ2 .
Suppose we want to deduce the value of the roots from b “ ´pξ1 ` ξ2 q. We clearly
need more information than the value of b alone and, in particular, it would be
useful to know s :“ ξ1 ´ ξ2 , since then
´b ` s
´b ´ s
and ξ2 “
;
ξ1 “
2
2
that is, the roots are given by
´b ˘ s
.
2
2
2
Observe that s “ pξ1 ´ ξ2 q is a symmetric polynomial in ξ1 , ξ2 and so by the fundamental theorem of symmetric polynomials there must exist some Q P CrX1 , X2 s
such that
s2 “ pξ1 ´ ξ2 q2 “ Qpb, cq.
COURSE OVERVIEW
3
Indeed, we can take QpX1 , X2 q :“ X12 ´ 4X2 since
pξ1 ´ ξ2 q2 “ pξ1 ` ξ2 q2 ´ 4ξ1 ξ2 .
Combining these observations we see the roots are given by
a
?
´b ˘ Qpb, cq
´b ˘ b2 ´ 4c
“
,
2
2
as required.
Similar, but more involved, methods can be used to deduce the formulae for
roots of cubic and quintic equations.
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]