ABSTRACT
We observed that one feature that distinguishes the study of field extensions from that of arbitrary ring extensions is the fact that a field can be considered as a vector space over any of its subfields, an observation which makes the power of linear algebra available for the study of field extensions. A second characteristic feature comes to light when we consider – according to the general approach of studying with any algebraic structure also the mappings which preserve this structure – not just fields but also field homomorphisms. Namely, nonzero homomorphisms between fields (in contrast to homomorphisms between rings) are automatically injective and hence faithfully transfer the structure of the first field to the second field. Since one of the main motivations of the study of field extensions stems from the problem of finding the roots of polynomials, let us point out the role of field homomorphisms in this context.
