If $K$ is an extension field of $\mathbb{Q}$ such that $[K:\mathbb{Q}]=2$, prove that $K=\mathbb{Q}(\sqrt{d})$ for some square free integer $d$

Let $\alpha \in K - \mathbb{Q}$. Since $1 , α, α^2$ are linearly dependent over $\mathbb{Q}$, $aα^2 + bα + c = 0$, where $a, b, c \in \mathbb{Q}$ and not all of $a, b, c$ are zero. By multiplying a suitable nonzero integer, we can assume $a, b, c \in \mathbb{Z}$. If $a = 0$, we get $bα + c = 0$. Since $b$ or $c$ is not zero, this can't happen. Hence $a \neq 0$. Hence $\mathbb{Q}(\alpha) = \mathbb{Q}((b^2 - 4ac)^{1/2})$. Since 1$ , α$ are linearly independent over $\mathbb{Q}$, [$\mathbb{Q}(\alpha) : \mathbb{Q}$] = 2. Hence $K = \mathbb{Q}(\alpha)$ and we are done.

This is easy enough once you know the primitive element theorem. Since any finite extension of $\Bbb{Q}$ is separable it follows that $K = \Bbb{Q}(\alpha)$ for some $\alpha\notin \Bbb{Q}$ that has degree 2 over $\Bbb{Q}$. Suppose without loss of generality that the minimal polynomial of $\alpha$ over $\Bbb{Q}$ is

$$x^2 + bx + c.$$

Then by the quadratic formula, $\alpha = \frac{-b\pm \sqrt{b^2 - 4c}}{2}.$ Now $b^2 - 4c$ cannot be a square in $\Bbb{Q}$ for then this would contradict $\alpha$ not being a rational number. Furthermore because $b$ is a rational number we have

$$\Bbb{Q}(\alpha)= \Bbb{Q}(\sqrt{b^2 -4c})$$

so finally at the end we can say that $K = \Bbb{Q}(\sqrt{e})$ where $e = b^2 - 4c$ that is not a square in $\Bbb{Q}$. Now write $e = \alpha/\beta$ where $\alpha,\beta$ are integers with $\alpha,\beta \neq 0$. We may assume that $e$ is in its lowest terms otherwise we can cancel factors off top and bottom. Then

$$\sqrt{e} = \sqrt{\alpha}/\sqrt{\beta} = \sqrt{\alpha\beta}/\beta$$

and consequently $K = \Bbb{Q}(\sqrt{e}) = \Bbb{Q}(\sqrt{\alpha\beta})$ because $\beta$ is an integer. Now if we write $d = \alpha\beta$, we have shown that $K = \Bbb{Q}(\sqrt{d})$, where $d$ is not the square of any integer.

$$\hspace{5in} \square$$