Let $K \subset L$ be an algebraic extension of fields finitely presented over a prime field or over an algebraically closed field. Is there an efficient procedure to check that $L/K$ is Galois? To compute the minimal normal extension $K \subset L'$ containing L?



efficiently checking that a field extension is Galois
I assume that you have given your field extension $L=K(\alpha\_1,\ldots,\alpha\_n)$ in the form of a list of polynomials generating the kernel $I$ of $K[X\_1,\ldots,X\_n]\twoheadrightarrow L$.
The intersection of this kernel with $K[X\_i]$ is computable (Gröbner basis wrt an elimination ordering) and gives you the minimal polynomial $p\_i$ of $\alpha\_i$ over $K$. $K(\alpha\_i)$ is separable over $K$ iff the derivative of $p\_i$ does not vanish. Hence separability of $L|K$ is decidable.


The extension is normal iff there are enough roots of the $p\_i$, i.e. iff there are $\deg(p\_i)$ roots of $p\_i$ in $L$ (which must be pairwise distinct to have a separable extension but we've already covered separability) and the minimal normal extension containing $L$ is exactly obtained by formally adjoining all missing roots. This seems to me to be equivalent to the problem of factorizing polynomials into irreducibles over finite extensions of $K$. Certainly if you can factorize polynomials (which is the case if $K$ is finite or $K=\mathbb{Q}$ but I'm not sure about more general fields), then you can check normality by testing if each $p\_i$ completely split into linear factors and you can compute the normal closure by recursively adjoining formal zeros until all $p\_i$ completely split.


