Smallest Field containing Subfield and Complex Number/Examples/Field Containing Rationals, Root 2 and Root 3

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 $\Q \sqbrk {\sqrt 2}$ does not contain $\sqrt 3$.

Proof
Let us denote $\Q \sqbrk {\sqrt 2}$ by $K$

Let $K'$ be a field containing $K$ and $\sqrt 3$.

Then $K'$ must contain all real numbers of the form:
 * $\paren {a + b \sqrt 2} \paren {c + d \sqrt 2} \sqrt 3$

where $a, b, c, d \in \Q$.

This can be written:
 * $\set {a + b \sqrt 2 + c \sqrt 3 + d \sqrt 6}$

and real numbers of this form do actually form a field.

Let $L := \Q \sqbrk {\sqrt 3}$ denote the set:
 * $L := \set {a + b \sqrt 3: a, b \in \Q}$

that is, all numbers of the form $a + b \sqrt 3$ where $a$ and $b$ are rational numbers.

Let $L'$ be a field containing $L$ and $\sqrt 2$.

Then $L'$ must contain all real numbers of the form:
 * $\paren {a + b \sqrt 3} \paren {c + d \sqrt 3} \sqrt 2$

where $a, b, c, d \in \Q$.

This can be written:
 * $\set {a + b \sqrt 3 + c \sqrt 2 + d \sqrt 6}$

which is the same set as before.

That is:
 * $K \sqbrk {\sqrt 2} = L \sqbrk {\sqrt 3}$

But a field containing $\sqrt 2$ and $\sqrt 3$ must also contain $\sqrt 6$.

Thus such a field contains all numbers of the form $a + b \sqrt 2 + c \sqrt 3 + d \sqrt 6$.

As these form a field, this is the smallest field containing $\Q$, $\sqrt 2$ and $\sqrt 3$.

Let this be denoted $\Q \sqbrk {\sqrt 2, \sqrt 3}$.

So:
 * $\Q \sqbrk {\sqrt 2, \sqrt 3} = K \sqbrk {\sqrt 3}$

We have that:
 * $\index {K \sqrt 3} K = 2$

and:


 * $\index K Q = 2$

and so by Degree of Field Extensions is Multiplicative:
 * $\index {\Q \sqbrk {\sqrt 2, \sqrt 3} } \Q = 4$

and it is seen that $\set {1, \sqrt 2, \sqrt 3, \sqrt 6}$ forms a basis for $\Q \sqbrk {\sqrt 2, \sqrt 3}$.