Smallest Field containing Subfield and Complex Number/Examples
Jump to navigation
Jump to search
Examples of Smallest Field containing Subfield and Complex Number
Complex Numbers
The field of complex numbers is the smallest field containing $\R$ and $i$.
Numbers of Type $a + b \sqrt 2: a, b \in \Q$
Let $\Q \sqbrk {\sqrt 2}$ denote the set:
- $\Q \sqbrk {\sqrt 2} := \set {a + b \sqrt 2: a, b \in \Q}$
that is, all numbers of the form $a + b \sqrt 2$ where $a$ and $b$ are rational numbers.
Then $\Q \sqbrk {\sqrt 2}$ is the smallest field containing $\Q$ and $\sqrt 2$.
Smallest Field Containing $\Q$, $\sqrt 2$ and $\sqrt 3$
The smallest field containing $\Q$, $\sqrt 2$ and $\sqrt 3$ is:
- $\set {a + b \sqrt 2 + c \sqrt 3 + d \sqrt 6: a, b, c, d \in \Q}$
This forms a vector space of dimension $4$ which has basis $\set {1, \sqrt 2, \sqrt 3, \sqrt 6}$.
Numbers of Type $a + b \sqrt [3] 2: a, b \in \Q$
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:
- $\index {\Q \sqbrk {\sqrt [3] 2} } \Q = 3$