Matrix fields, regular and irregular: A complete fundamental characterization (Q1084478)

Let F be a field and \((F)_ n\) the algebra of \(n\times n\) matrices over F. Let p (zero or prime) be the characteristic of F and let \(F_ p\) be its prime subfield. It is obvious that every extension field K of \(F_ p\) with [K: \(K\cap F]\leq n\) can be embedded in \((F)_ n\). This paper deals with the nature of the embeddings of fields in \((F)_ n\) in the case where [F: \(F_ p]\) is finite. Let K be a subfield of \((F)_ n\). All elements of K have the same rank, m say. Using a similarity we may diagonalize the identity of K and hence K becomes similar to \(K_ 0\oplus 0_{n-m}\) where \(K_ 0\subseteq (F)_ m\) contains the identity \(m\times m\) matrix. Thus, in classifying K, we may assume \(m=n\). Then K contains the scalar matrices \(F_ pI\). Also \(K=F_ p(A)\) for some \(n\times n\) matrix A over F. As a generator of K over \(F_ p\), A has a minimal polynomial \(g(x)\in F_ p[x]\). (This is also the minimal polynomial of A regarded as a matrix over \(F_ p\) where F is represented in the regular way over \(F_ p)\) and as an element of \((F)_ n\) it has a minimal polynomial f(x). The relationship between f(x) and g(x) is explored in this paper. Related to this is the problem of describing all the scalar matrices in K. Let \(S_ n(K)=\{a\in F|\) aI\(\in K\}\). Let \(F_ f\) be the field generated over \(F_ p\) by the coefficients of f. It is clear that \(S_ n(K)\subseteq F_ f\) but equality may not hold. The quite subtle connection between \(S_ n(K)\) and \(F_ f\) is studied, particularly in the case where K is normal over \(F_ p\). The paper contains nice results linking this problem to the Galois action on the prime factors of g(x) over F.