
22 Classical Algebra
so
(p + q
√
2)
−1
=
p
p
2
−2q
2
−
q
p
2
−2q
2
√
2
if p and q are non-zero. The map p + q
√
2 7→ p −q
√
2 is an automorphism of K.
(3) Let α =
3
√
2 ∈ R, and let
ω = −
1
2
+ i
√
3
2
be a primitive cube root of unity in C. The set of all numbers p + qα + rα
2
, for
p,q,r ∈ Q, is a subfield of C, see Exercise 1.5. The map
p + qα + rα
2
7→ p + qωα + rω
2
α
2
is a monomorphism onto its image, but not an automorphism, Exercise 1.6.
1.3 Solving Equations
A physicist friend of mine once complained that while every physicist knew what
the big problems of physics were, his mathematical colleagues never seemed to be
able to tell him what the big problems of mathematics were. It took ...