Smallest Field containing Subfield and Complex Number/Examples/Numbers of Type Rational a plus b cube root 2

Example of Smallest Field containing Subfield and Complex Number
Let $\Q \sqbrk {\sqrt [3] 2}$ denote the set:
 * $\Q \sqbrk {\sqrt [3] 2} := \set {a + b \sqrt [3] 2 + c \sqrt [3] {2^2}: a, b, c \in \Q}$

Then:
 * $\Q \sqbrk {\sqrt [3] 2}$ is the smallest field containing $\Q$ and $\sqrt [3] 2$

and:
 * $\index {\Q \sqbrk {\sqrt [3] 2} } \Q = 3$