Review of Fields (Algebra 3: Lecture 7 Video 1)

Lecture 7: We started this lecture by recalling some examples of fields that we mentioned during our discussion of rings at the end of Math 206A.  We then recalled a result about quotients of polynomial rings by ideals.  We reviewed the field of fractions of an integral domain.  We recalled the definition of the characteristic of a ring and some of its basic properties, including the fact that the characteristic of a field is either a prime p, or is 0.  We saw how this led to the fact that a field F either contains a subfield isomorphic to F_p for some p or to Q.  We defined an extension of fields and saw that if K is an extension of F, then K is a vector space over F.  We saw some consequences for the special case of finite fields.  In the last video of the lecture we discussed extensions of fields and roots of polynomials.  We proved that if F is a field and p(x) is a nonconstant polynomial, then there exists an extension K of F in which p(x) has a root.  We proved a stronger statement, that there exists an extension L of F such that p(x) splits completely as a product of linear factors in L[x].  We defined a splitting field and proved that for any F and any p(x), there exists a splitting field for p(x).  We ended the lecture with some questions that we will return to in future lectures.