have solutions. Remarkably, Galois theory also provides clear explana-
tions of three compass-and-straightedge impossibilities we have previ-
ously examined: why the cube cannot be duplicated, why the angle cannot
be trisected, and why only certain regular polygons are constructable.Galois Groups
When I first learned the quadratic formula in high school, my algebra
teacher mentioned that there were such formulas for polynomials of degree
three and degree four, but no such formula existed for polynomials of de-
gree five. At the time, I didn’t completely understand what the teacher
meant, and interpreted his remark to mean that mathematicians simply
hadn’t discovered the formula yet. It wasn’t until later that I realized that
although there were formulas that did give the roots of fifth-degree polyno-
mials, those formulas used expressions other than radicals—if the “lan-
guage” for describing solutions consisted simply of whole numbers,
radicals, and algebraic expressions involving them, then that language sim-
ply doesn’t have a means of expressing the roots of all fifth-degree polyno-
mials. One of my goals as a student was to find out why this was so—but in
order to fully understand it, one must learn Galois theory. In order to un-
derstand Galois theory, it is necessary first to take an introductory course in
abstract algebra, which usually comes about the third year of college.
Nonetheless, it is possible to understand some of the basic ideas sur-
rounding the theory. Using the quadratic formula, the polynomial
x^2 6 x4 has two roots: A 3 �5 and B 3 �5. These roots satisfy two
basic algebraic equations: AB 6 and AB4. Admittedly, they satisfy
a whole bunch more, such as 5(AB)3(AB)^3 5 6 3 64 162, but
this equation was obviously constructed from the other two. They also
satisfy AB 2 �5, but this equation is qualitatively different from the
first two: the only numbers that appear in the first two equations are ra-
tional numbers, whereas the last equation contains an irrational number.
Notice also that if we tried something like A 2 B, we would get the irra-
tional number 9� 5 , so the equations that can be constructed from A
and B that involve only rational numbers are definitely limited.
Look once again at the two equations AB 6 and AB4, but instead of
writing them in this form, write them in the form �� 6 and ��4,
where the plan is to look at the various possible ways of inserting the two
roots A and B into the � and � locations in order to get a true statement.
There are two ways that this can be done. One is the original way we ob-
tained these equations—insert A into the � and B into the �, which
The Hope Diamond of Mathematics 95