Field Extension/Examples/Numbers of Type Rational a plus b root 2

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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 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