Smallest Field containing Subfield and Complex Number/Examples/Numbers of Type Rational a plus b root 2

Example of Smallest Field containing Subfield and Complex Number

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 $\sqrt 2$.

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Formally, $\Q \sqbrk {\sqrt 2}$ is the field extension of $\Q$ for the minimal polynomial of $\sqrt 2$, the second-degree polynomial $x^2 - 2$.

Therefore, $\Q \sqbrk {\sqrt 2}$ is the vector space of dimension $2$ isomorphic to $\Q \sqbrk x / \gen {x^2 - 2}$.

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 $72$
• 2017: Joseph A. Gallian: Contemporary Abstract Algebra (9th ed.) ... (previous) ... (next): Chapter $20$: Extension Fields: $\S 1$. Splitting Fields