Math 366 Final Exam review topics (Spring 2010)
This exam will not be intentionally cumulative, though rings contain groups, so knowledge from
groups could conceivably be beneficial here and there.
Exam 1 covered groups; this one covers rings and fields (no applications/advanced topics). For
general study tips for my exams, please take a look at the review sheet for Exam 1. Here are some
things to know for this exam:
1. Basic ideas about rings
(a) Be able to show that something is a ring (10 things to check) or explain why it isn’t.
(b) What makes a ring commutative and/or a ring with unity?
(c) What are subrings and ideals? What’s the difference?
(d) What do you get if you mod a ring by an ideal? What kind of elements do you get?
(maybe think about some examples)
(e) What makes a map between rings a ring homomorphism? Isomorphism/automorphism?
What is the kernel of a homomorphism, and what does the first isomorphism theorem
say?
2. Special rings
(a) What is a zero divisor?
(b) What are integral domains and fields? Can you think of a ring that isn’t an integral
domain or an integral domain that isn’t a field?
(c) What is the characteristic of a ring? If I hand you a ring, how can you compute it?
(d) What is a polynomial ring? What are the elements of a polynomial ring? What if you
mod out by an ideal?
(e) What does the Fundamental Theorem of Algebra say?
(f) How do you check whether a polynomial is irreducible? For that matter, what does it
mean to be irreducible? What does it mean to factor a polynomial over some field?
(g) When is F [x]/ < p(x) > a field?
(h) There won’t be anything about UFDs, Fermat’s last theorem, Euclidean domains, or
PIDs.
3. Fields
(a) What is a vector space? What is a basis? What is the dimension of a vector space? (369
and 466/467 cover vector spaces in much more detail, so I won’t expect much from that
section.)
(b) What is a splitting field? What does it mean for a polynomial to split over a field?
(c) We had an arcane way of building a splitting field (by moving to the quotient of a
polynomial ring by a certain ideal). How does that go? What are the elements of the
splitting field of pitched this way?
(d) What is the more natural way of writing down a splitting field? What are the elements
of the splitting field if pitched this way?
(e) What makes a field extension algebraic? Transcendental? What is the degree of an
algebraic field extension?
(f) What orders are possible for finite fields? What subfields does a finite field have?