Let $L/K$ be an arbitrary algebraic field extension. How is a normal closure of $L$ (the smallest normal extension of $K$ containing $L$) constructed?
If $L/K$ is finite, then writing $L=K(\alpha_1,...,\alpha_n)$ the splitting field of the product of the minimum polynomials $\prod_i m_{K,\alpha_i}(X)$ is such a field, unique up to isomorphism.
Is there a method for its construction in the general case using a Zorn's lemma type argument?
Many thanks.