Field Extension/Examples/Real Numbers of Type Rational a plus b root 2
Examples of Field Extensions
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 $2$.
Proof
From Real Numbers of Type Rational a plus b root 2 form Field, $\Q \sqbrk {\sqrt 2}$ forms a field.
From Rational Numbers form Field, $\Q$ is also a field.
We have that $\Q \subseteq \Q \sqbrk {\sqrt 2}$, as:
- $\Q = \set {x \in \Q \sqbrk {\sqrt 2}: b = 0}$
Thus $\Q \sqbrk {\sqrt 2}$ is a field extension of $\Q$.
Thus $\Q \sqbrk {\sqrt 2}$ can be considered as a vector space over $\Q$.
Then we have that $\set {1, \sqrt 2}$ forms a basis of $\Q \sqbrk {\sqrt 2}$.
Hence $\Q \sqbrk {\sqrt 2}$ forms a finite field extension over the rational numbers $\Q$ of degree $2$.
$\blacksquare$
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 $70$