Field Extension/Examples
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Examples of Field Extensions
Complex Numbers over Reals
The complex numbers $\C$ forms a finite field extension over the real numbers $\R$ of degree $2$.
Real 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 $\Q \sqbrk {\sqrt 2}$ forms a finite field extension over the rational numbers $\Q$ of degree $2$.