I've seen proofs of the above but I'd like to know if the following is also a valid proof:
So [K:Q] = 2 < $\infty$, so by the simple extension theorem: $\exists \theta \in K$ such that K = $\mathbb{Q}(\theta)$
Claim 1: $\mathbb{Q}(\theta)$ = $\mathbb{Q}(\theta + q) $ for all $q \in \mathbb{Q} -\{0\}$
Proof:
a + b$\theta \in \mathbb{Q}(\theta) \Rightarrow a-bq+bq+b\theta = a-bq + b(q+\theta) \in \mathbb{Q}(\theta + q) \Rightarrow \mathbb{Q}(\theta)\subseteq \mathbb{Q}(\theta + q)$
a+b($\theta+q) \in \mathbb{Q}(\theta + q) \Rightarrow a+bq + b\theta \in \mathbb{Q}(\theta) \Rightarrow \mathbb{Q}(\theta+q)\subseteq \mathbb{Q}(\theta)$
So, $\mathbb{Q}(\theta) = \mathbb{Q}(\theta + q)$
$\blacksquare$
Claim 2: $\mathbb{Q}(\theta)$ = $\mathbb{Q}(q\theta) $ for all $q \in \mathbb{Q} -\{1\}$
Proof: similar to claim 1.
Then without a loss of generality: K = $\mathbb{Q}(\theta)$ with $\theta$ such that: $\theta = q_1\phi + q_2$, $\phi \in K, q_1, q_2 \in \mathbb{Q} \iff q_1 = 1$ and $q_2 = 0$
Then since [K:Q] = 2, $\theta$ is the root of a minimal polynomial of degree two: $P_{\theta}(x) = x^2 + \alpha x - \beta$, $\alpha, \beta \in \mathbb{Q}$
Clearly $\beta \neq 0$, otherwise the polynomial is not irreducible. Since $\theta$ is a root: $\theta^2 + \alpha \theta - \beta = 0 \Rightarrow \theta = \dfrac{-\alpha}{2} \pm \dfrac{\sqrt{\alpha^2 + 4\beta}}{2}$
But since this contradicts my assumption about theta not being written as a linear combination with coefficients in $\mathbb{Q}$, $\alpha = 0$
so $P_\theta(x) = x^2 - \beta$, but this is irreducible over $\mathbb{Q}$ if and only if $\sqrt{\beta}$ is not rational.
Moreover since $\sqrt{\beta} = \sqrt{\dfrac{\beta_1}{\beta_2}} = \dfrac{\sqrt{\beta_1\beta_2}}{\beta_2} $, $\beta_1, \beta_2 \in \mathbb{Z}$
But my claim about the form of $\theta$ requires that $\beta_2 = 1$ and my claim about the form of $\theta$ will mean that $\beta_1$ is square free (as otherwise there is a rational coefficient)
so $\theta = \sqrt{\beta_1}$
Is this proof fundamentally flawed by my assumption about the form of theta and if so why?
Thanks!