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:

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

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 36$. The Degree of a Field Extension: Example $74$