MA 1560: Number Theory, Review exercises

These exercises are meant as review of MATH 1530. You do not have to hand them in, but I strongly encourage you to do them! We will use this material in MATH 1560, and I will assume you are `fluent' in these exercises.

It is no problem if you get stuck at these exercises at first, that is normal after a snowy Christmas break! Feel free to consult an algebra textbook, digest the proofs there and try to write them up in your own words (it helps if you put the book away at this point).

Problem A.1
Let G be a finite group, let H be a subgroup of G, and let g ∈ G be an element of G.
(a) Recall that the order of G is the number of elements in G, and similarly for H. What is the definition of the order of the element g?
(b) Lagrange's Theorem (Part 1): Prove that the order of H divides the order of G
(c) Lagrange's Theorem (Part 2): Prove that the order of g divides the order of G

Problem A.2
Let R be a ring.
(a) What is the definition of an ideal?
(b) What is the definition of a maximal ideal?
(c) Let I be an ideal of R. Prove that the quotient ring R/I is a field if and only if I is a maximal ideal. (You may assume that R/I is already a ring, so what you need to prove is that every non-zero element of R/I has an inverse if and only if I is a maximal ideal.)

Problem A.3

Let Z be the ring of integers {...-2,-1,0,1,2,...}.
(a) Prove that every ideal in Z is principal, i.e., generated by a single element.
(b) Let m ∈ Z. The ideal generated by m is denoted by (m) or mZ. Prove that mZ is a maximal ideal if and only if |m| is prime. Deduce that the quotient ring Z/mZ is a field if and only if |m| is prime.

Problem A.4

Let F be a field.
(a) Let f(x) ∈ F[x] be a non-zero polynomial of degree d. Prove that f(x) has at most d distinct roots in F.
(b) Give an example of a ring R and a non-zero polynomial f(x) ∈ R[x] of degree d that has more than d roots in R.
(c) Let F be a finite field, i.e., a field with finitely many elements. Prove that there is a prime p such that pa = 0 for all a ∈ F. (Here pa means to add a to itself p times.)
(d) Let F be a finite field. Prove that the number of elements in F is a power of a prime. (Hint. Let p be the prime from (c), and prove that F is a finite dimensional vector space over the field Z/pZ.