The Embedding Problem in Galois Theory

The central problem of modern Galois theory involves the inverse problem: given a field $k$ and a group $G$, construct an extension $L/k$ with Galois group $G$. This book features lectures on embedding theory presented at St Petersburg University and contains the main results on the embedding problem. It presents various stages of development.

The central problem of modern Galois theory involves the inverse problem: given a field $k$ and a group $G$, construct an extension $L/k$ with Galois group $G$. The embedding problem for fields generalizes the inverse problem and consists in finding the conditions under which one can construct a field $L$ normal over $k$, with group $G$, such that $L$ extends a given normal extension $K/k$ with Galois group $G/A$. Moreover, the requirements applied to the object $L$ to be found are usually weakened: it is not necessary for $L$ to be a field, but $L$ must be a Galois algebra over the field $k$, with group $G$.In this setting, the embedding problem is rich in content. But the inverse problem in terms of Galois algebras is poor in content because a Galois algebra providing a solution of the inverse problem always exists and may be easily constructed. The embedding problem is a fruitful approach to the solution of the inverse problem in Galois theory. This book is based on D. K. Faddeev''s lectures on embedding theory at St. Petersburg University and contains the main results on the embedding problem. All stages of development are presented in a methodical and unified manner.