Homework Number Seven

This is due on Wed Apr 23.

Start by recalling a few definitions.

• A group is a set G equipped with an associative binary operation which has an identity element and is such that every element has an inverse.
• A ring (in this course) is a set R equipped with two assocative and commutative operations + and x such that R is a group under +, x has an identity, and x distributes over +. The +-identity is called 0 and the x-identity is called 1.
• A field is a ring in which 1 is not equal to 0, and every nonzero element has a x-inverse.

 

1. Recall that if E is a field then E[x] is the ring of polynomials with coefficients from E. As we stated in class E[x] has a reasonable notion of division: given f,g in E[x] with g nonzero there exist unique q,r in E[x] such that f = q g + r and r=0 or deg(r) < deg(g). In this case we write r = f mod g and call it the *remainder*.
   1. Divide x^3 + 2 x + 1 by x - 2 in the ring of polynomials with real coefficients.
   2. Let E=Z/3 Z. In a mild abuse of notation we write E = { 0, 1, 2 }, that is we let each of the numbers 0,1,2 stand for the corresponding congruence class. Divide x^3 + 2 x + 1 by x -2 in the ring E[x]
2. Recall that if E is a field then we say f in E[x] is *irreducible* in E[x] when deg(f) > 0, and f = g h implies that deg(g) = 0 or deg(h) = 0 for all g, h in E[x].
   1. Prove that if E=Z/2 Z then each of the polynomials x^3 + x + 1, x^3 + x^2 + 1 is irreducible in E[x].
   2. Is x^4 + x^2 + 1 an irreducible polynomial in the ring of polynomials with real coefficients?
3. As we saw in class, when f in E[x] is irreducible in E[x] and deg(f) = n we can construct a field as follows: let P be the set of polynomials in E[x] of form c_0 + c_1 x + ..... + c_{n-1} x^{n-1} with c_i in E and let the field operations be addition modulo f and multiplication modulo f.
   1. Let E= Z/2 Z and f = x^3 + x + 1. We can perform the construction we described above to get a field F. What is the multiplicative inverse of x + 1? What is the least n > 0 such that (x+1)^n = 1 ?
   2. Let E= Z/2 Z and g = x^3 + x^2 + 1. We can perform the construction we described above to get a field G. What is the multiplicative inverse of x + 1? What is the least n > 0 such that (x+1)^n = 1 ?
   3. (Harder) Prove that F and G are *isomorphic*: that is there is a permutation phi of P such that phi(a +_F b) = phi(a) +_G phi(b) for all a, b in P and similarly for multiplication. Here I am writing +_F for the addition operation of F which is addition modulo f, and +_G for the addition operation of G which is addition modulo g.
4. Let E be a field. Define f(n) to be the result of forming the sum of n 1's in E. Prove that f(a b) = f(a) f(b) and f(a + b) = f(a) + f(b). Prove that if f(n)=0 for some n > 0 then the least such n is prime.