Math 314  Homework Assignment 1  January 15, 2004

due date: January 29, 2004

1. Suppose that F is a field and B a ring, and ϕ : F ⟶ B a ring homomorphism. Show that ϕ is either an injection or the zero map. (If you use the convention that for ring homomorphisms, 1_{F} has to get sent to 1_{B}, then this implies that all ring homomorphisms ϕ : F ⟶ B have to be injections, unless B is the zero ring.)

Is the same thing true for ring homomorphisms ϕ : B ⟶ F?

2. Suppose that A is a ring, and f_{1},…, f_{n} any elements of A. Define the set

(f_{1}, …, f_{n}) =^{def} {a_{1}f_{1} + a_{2}f_{2} + ⋯ a_{n}f_{n} a_{1}, …, a_{n} ∈ A} ⊆ A

to be the set of “linear combinations” of f_{1}, …, f_{n} with coefficients in A. i.e., the set of all possible sums of the above form.

Show that this is an ideal of A.

3. Suppose that A is a ring with infinitely many elements with the following property. For every nonzero ideal I of A, the quotient ring A/I is finite (that is, it has only finitely many elements). Show that A is a domain.

Note 1: Such rings exist: A = ℤ is an example.

Note 2: As always, we’re assuming that our rings contain a 1. (The way that the “1” appears is slightly subtle and easy to overlook. So, if you have a solution, but it doesn’t explicitly seem to use that fact that there’s a 1 in the ring, don’t worry about it, it doesn’t necessarily mean that your solution is wrong).

Note 3: This problem is completely useless in practice. I have never seen a situation where you would apply the fact/lemma that you’re supposed to prove above. It is very good for one thing though: you can’t solve it without having a good picture of how quotient rings and ideals work.

4. Let A be the ring ℚ[x], f the polynomial f = x^{3} + 3x^{2} + 3x − 1, and I the ideal of A consisting of all multiples of f (i.e. I = (f) in the notation of question 2). Find representatives of degree < 3 in the ring B = A/I for the following expressions:

(a) x^{4}.

(b) (x^{2} + 1)^{3}.

(c) 1/x.

5. Let α be the real number α = 2^{1/3} − 1. To as many decimal places as you can (well, at least 6, and no more than 20), evaluate the following real numbers:

(a) α^{4}.

(b) (α^{2} + 1)^{3}.

(c) 1/α.

(d) α^{2} + 3α + 3.

(e) 24α^{2} + 60α − 16.

(f) 6α^{2} + 10α − 3.

6. A field F with only finitely many elements is called a finite field. We know some (but not all) examples of these: for any prime number p, the field 𝔽_{p} has exactly p elements.

Suppose that F is a finite field. Explain why char(F) cannot be zero, and so must be some prime number p.

If p = char(F), show that the number of elements in F is a power of p. This shows, for example, that there are no finite fields with 6 elements. Do you think it is possible to have a finite field with 4 elements?