
32 Classical Algebra
1.4* Prove without using Cardano’s formula that
3
q
18 +
√
325 +
3
q
18 −
√
325 = 3
1.5 Let α =
3
√
2 ∈R. Prove that the set of all numbers p+qα +rα
2
, for p,q, r ∈Q,
is a subfield of C.
1.6 Let ω be a primitive cube root of unity in C. With the notation of Exercise 1.5,
show that the map
p + qα + rα
2
7→ p + qωα + rω
2
α
2
is a monomorphism onto its image, but not an automorphism.
1.7 Use Bombelli’s observation that (2 ±
√
−1)
3
= 2 ±
√
−121 to show that (with
one choice of values of the cube roots)
3
q
2 +
√
−121 +
3
q
2 −
√
−121 = 4
1.8 Use the identity cos 3θ = 4 cos
3
θ −3 cos θ to solve the cubic equation t
3
+
pt + q = 0 when 27q
2
+ 4p
3
< 0.
1.9 Find radical expressions ...