gives the original two (true) statements AB 6 and AB4. If a deck of
two cards is shuff led, where A is initially on top and B is initially on the
bottom, then � represents the letter that ends on top and � the letter that
ends on the bottom after the shuff le. The substitution A for � and B for
� corresponds to the phantom shuff le. The only other shuff le of two
cards has A ending on the bottom and B on top, so when B is substituted
for � and A for �, the resulting statements BA 6 and BA4 are still
true. The Galois group of a polynomial consists of all those shuff les that
result in all the algebraic equations with rational numbers being true
statements. So the Galois group of the polynomial x^2 6 x4 consists of
the two shuff les (phantom and switch top-two cards) that comprise S 2.
It is not always the case that both shuff les in S 2 are in the Galois group
of the polynomial. To see such a case, consider the polynomial x^2 2 x3,
whose two roots A and B are 3 and 1. The two roots satisfy A 2 B 1, so
examine the algebraic equation with rational coefficients � 2 � 1. If A
and B are switched in the left side of the equation, the resulting equation
is B 2 A1, which is not a true statement, as the sum B 2 A actually
equals 5. For this polynomial, the only shuff le that generates true state-
ments from the original equations is the identity element (the phantom
shuff le), so in this case the Galois group of x^2 2 x3 consists of just the
phantom shuff le.
There is a famous quote from the American astronomer Nathaniel Bow-
ditch, who translated Laplace’s Celestial Mechanics into English. Bowditch
remarked, “I never come across one of Laplace’s ‘Thus it plainly appears’
without feeling sure that I have hours of hard work before me to fill up the
chasm and find out and show how it plainly appears.”^11 The same is gener-
ally true for the statement “It can be shown,” so I am loathe to include it
unless I absolutely must—but here I absolutely must. It can be shown that
a polynomial has roots that can be expressed in terms of radicals only
when its Galois group has a particular structure in terms of its subgroups.
This structure is known as solvability; it is quite technical to describe, but
the name is clearly motivated by the problem of solving the problem of
finding the roots of a polynomial by radicals. The Galois group of the poly-
nomial x^5 x1 can be shown (oops, I did it again) not to be solvable, and
so the roots of that polynomial cannot be found by radicals.
Later Developments
The insolvability of the quintic proved to be a significant moment in the
development of mathematics. It is not possible to say with certainty what
would have happened had quintics, and polynomials of higher degree,
96 How Math Explains the World