Describe the embeddings of L extending σ1, and the embeddings of L extending σ2. Which are real and which are complex? What is the signature of L?

Question

Let $K = \Bbb{Q}(\sqrt2)$ and let $\alpha = 1 + \sqrt2.$ Let $\beta = \sqrt\alpha$ and $L = \Bbb{Q}(\beta).$ I've already shown that the embeddings of K can be labelled as $\sigma_1, \sigma_2$ in such a way that $\sigma_1(\alpha) > 0$ and $\sigma_2(\alpha) < 0$; and $\beta\notin K$. How do I describe the embeddings of $L$ extending $\sigma_1$ and $\sigma_2$? How do I even know that $L/K$ is an extension of $L$?