Degree of a field extension is a power of 2

My question is somewhat similar to Dimension of a Finite Field Extension is a Power of 2 but there's a small complication.

Claim 1: Let $K$ be a field of characteristic $\neq 2$, and suppose that every polynomial of odd degree in $K[x]$ has a root in $K$. Let $L/K$ be a finite extension. Then $[L:K] = 2^n$ for some $n$.

Claim 2: Let $K,L$ be as above. If $L = L^2$, then $L$ is algebraically closed.

For Claim 1, I have the following reasoning, but it doesn't quite get me all the way there. Since every odd degree polynomial has a root, it is reducible into a linear factor and even degree polynomial, so all irreducible polynomials are even degree. Since $L/K$ is finite, it is finitely generated, so $L = K(\alpha_1,\ldots, \alpha_n)$ for some $\alpha_1,\ldots, \alpha_n \in L$. Then we have a tower of fields
$$
K \subset K(\alpha_1) \subset K(\alpha_1,\alpha_2) \subset \ldots \subset K(\alpha_1,\ldots, \alpha_n) = L
$$
In each step of the tower, $K(\alpha_1, \ldots, \alpha_{i+1}) = K(\alpha_1, \ldots, \alpha_i)$ and the irreducible polynomial of $\alpha_{i+1}$ is even degree, so each extension is even. Thus $[L:K]$ is even.

If I could show that each step of the tower is of degree 2, then I would have proved claim 1, but I don't know why that would be the case. Is there a reason that we can't have irreducible quartics over $K$?