Still in Chapter 2 - moving from rings to fields.

Many of you commented in your first homework that you were pleasantly surprised to find Sethuraman's style so conversational. Then when you set out to write your own mathematics, you tried to be as far from conversational as possible. Not a good plan. From now on when you write your homework you should consciously try for a conversational style. Use short sentences. Explain what you are trying to say, in words. Use algebra sparingly.

There's probably way too much work here for most of you. So don't waste precious time on problems you are totally stumped by. Work a little on each one to see which you think you can manage, and then polish those off before you struggle with the ones that are harder (for you).

1. In class we conjectured that Z_n is a field just when n is prime. Prove that. Let me remind you that the "just when" means you have two things to prove: first that when n is not prime, Z_n is not a field, and second that when n is prime every nonzero element in Z_n has a multiplicative inverse and hence Z_n is a field.

   Optional extra challenge: prove that any finite integral domain is a field.

2. On pages 37 and 38 (and then again on page 60) Sethuraman discusses the ring of polynomials with real coefficients. Then he notes that there is nothing special (as far as ring theory goes) about real coefficients - the coefficients can come from any ring. We covered this in class. We write R[x] for the ring whose elements are the polynomials in the variable x whose coefficients come from the ring R.

   We know about the degree of a polynomial. In Z[x] the polynomial x²+1 has degree 2. The constant polynomial 37x⁰ has degree 0. In fact any nonzero constant polynomial has degree 0. What about the degree of the 0 polynomial? It's not degree 0, since the degree is supposed to be the largest exponent with a nonzero coefficient, but for the 0 polynomial there are no such exponents. There's a good reason for declaring that the degree of the 0 polynomial is -infinity (that's minus infinity. Take my word for that now.

   Let's count polynomials. Consider Z₂[x]. That's the ring of polynomials whose coefficients are just 0 and 1, with the usual addition and multiplication rules except for 1+1, which is 0 and not 2. In that ring there are four polynomials whose degree is at most 1, namely 0, 1, x and 1+x.

   1. How many polynomials are there in Z₂[x] with degree at most 2? at most 3? at most d?

   2. Answer the same question for Z₃[x], and then for Z_n[x] (if you can). That is, try to find a formula involving n and d that tells you the number of polynomials in Z_n[x] whose degree is at most d.

   3. Explain why there are exactly as many polynomials in Z_n[x] of degree at most d as there are words of length d+1 you can write using an alphabet with just n letters in it. (Words don't need to make any sense: "etaionshrdlu" is a word of length 12 using letters from our ordinary 26 letter alphabet.) (Where do you think I got that word?)

      "Explain why" is a gentle way to say "prove". It invites you to write a convincing argument, in English, with only the symbols you really need.

3. One of the hardest things an algebra teacher needs to do is to convince his or her students that (1+x)2 is not 1 + x2.

   Prove that

   	in Z2[x]: (1+x)2 = 1 + x2
   	in Z3[x]: (1+x)3 = 1 + x3
   

   What happens in Z₄[x] and Z₅[x]? Try to find a pattern, frame a conjecture, and prove it if you can.

4. Suppose f and g are polynomials in R[x]. Investigate the question

   	degree( fg ) = degree(f) + degree(g)
   

   Find out when it's true, and prove it when it's true.

   Can you say now why mathematicians define the degree of the 0 polynomial to be -infinity?

5. When you studied calculus you studied power series: expressions like these:

   	g(x) = 1 +  x +  x2 +  x3 + ...
   	h(x) = 1 + 2x + 3x2 + 4x3 + ...
   	f(x) = 1 +  x + 2x2 + 3x3 + 5x4 + 8x5 + ...
   
   

   Think of them as polynomials of infinite degree. When you first saw these you were probably asked to figure out when they converged. Here we won't bother! It's pretty clear how to add and to multiply power series, and that the coefficents don't need to be integers. They can come from any ring. This is an extremely powerful and interesting abstraction we can barely touch on, but it's so pretty I wanted you to see just a little bit of it. Here are some (easy?) exercises. Recall that if r is an element of a ring then we write (1/r) for the multiplicative inverse of r (if it has one). You can check whether some ring element s is equal to (1/r) by checking that r*s=1, since by definition (1/r) is the solution to the equation r*?=1

   1. The first series isn't called g by accident. It's the famous geometric series. Prove that

      	g(x)*(1-x) = 1
      

      so that (in any ring whatsoever) g(x) = 1/(1-x).

   2. The third power series above has the Fibonacci numbers as coefficients, which is why I called it f. Prove that f(x) = 1/(1-x-x2) .

   3. Hard (or at least tricky) question. Find the multiplicative inverse of the second power series, h(x) above. Hint: h seems to be the derivative of the geometric series.