The number of quadratic subfields

Abstract

We use elementary field theory to compute the number of intermediate fields of degree 2 of any finite extension of fields of characteristic not 2. This leads to necessary and sufficient conditions for such extensions to have no intermediate fields of degree 2, or to have a unique intermediate field of degree 2. We obtain several applications including the number of quadratic subfields of cyclotomic extensions.2010 Mathematics Subject Classification. 06D99, 08A30. Key words and phrases. Galois group, Galois extension, finite extension, intermediate fields, quadratic closure, cyclotomic extensions, subgroup of squares.Received on 22-1-2013; revised 25-2-2014. I would like to thank the referee for his/her comments, which improved the readability of the article.