Determine in each of the following cases whether the indicated set is a subring of C.

1. { x + y i 2 | x, y Z}.
2. { x + y a | x, y Z}, where a is the unique positive number such that a³ = 2.

Let R be a ring and let f : R -> R be a morphism. Prove that the subset S of R consisting of the elements x such that f(x) = x is a subring of R. What is this subring if f : C -> C is the map that sends any complex number to its complex conjugate?

Let a be an invertible element in the ring R. Show that the map f : R -> R defined by f(x) = axa^-1 is an isomorphism and determine its inverse. What happens if the ring R is commutative? For which elements a in the ring is the map g : R -> R defined by g(x) = axa a morphism?

Let S be a nonempty set and let R be a ring.

1. Show that the set Map(S, R) of maps from S to R is a ring, where the sum and product of two elements f and g are defined by

   (f + g)(s) = f(s) + g(s), (f * g)(s) = f(s) * g(s),

   and where the zero element (unit element) is the map that maps every s in S to 0 (1).
2. Let R = Z/2Z and for every subset A of S define the map f_A: S -> Z/2Z by f_A(a) = 1 if and only if a A. Show that the map F which sends the subset A of S to f_A is a bijection from the collection P(S) of all subsets of S to the set Map(S, Z/2Z) of maps from S to Z/2Z.
3. Use the bijection of the previous item to define a ring structure on P(S) and express the operations in terms of the familiar set theoretic operations like union and intersection.

Prove that the following maps are morphisms of rings.

1. : Q[X] -> Q, (f(X)) = f(2)     (substitution).
2. : Z[X] -> Z/nZ[X], given by reduction of the coefficients of polynomials modulo the integer n 2.

Let f : R -> S and g : S -> T be morphisms of rings.

1. Show that the composition g o f : R -> T is a morphism.
2. Show that Ker(g o f) = Ker(f) if g is an isomorphism.
3. Show that Ker(g) = f(Ker(g o f)) if f is an isomorphism.

Show that the map F : Z[X] ->Q defined by F(f) = f(1/2) is a morphism and determine its kernel and its image.

If we replace every occurrence of X in the polynomial f in Q[X] by aX + b, then the new polynomial is written as f(aX + b) and is again an element of Q[X]. Let F : Q[X] -> Q[X] be defined by F(f) = f(2X + 3). Show that F is an isomorphism and determine its inverse.

Prove the converse of the Cancellation law for domains. If for every nonzero element a R the implication

for all x, y R    ax = ay    implies   x = y,

holds, then R is a domain.

Show that the ring Q is not finitely generated.

Prove: If K is a field and : K -> K is an isomorphism, then

{x K | (x) = x}

is a subfield of K.

Prove that Q + Qd is a subfield of R for every positive integer d. When does it coincide with Q?

Let K be a field and let R be a subring of K. Note that R is then a domain. Prove that the map F : Q(R) -> K, given by F(a/b) = ab^-1, is a well-defined injective morphism and show that the image is the smallest subfield of K containing R. Conclude that Q(R) is isomorphic with the smallest subfield of K. Use this to describe the field of fractions of Z[X].

Suppose we were to introduce fractions of the ring S = Z/4Z, which is not a domain. Define the relation eaq as before. Does it define an equivalence relation on S?

In each of the following cases determine the ideal in the ring R generated by the set V:

1. V= {2} and R = Z/6Z;
2. V= {2,3} and R = Z/6Z;
3. V= {2} and R = Z/8Z;
4. V= {2,X} and R = Z[X].

Show that the following equality of ideals holds in Q[X,Y]:

Does it also hold in Z[X,Y]?

Determine if the set V is an ideal in the ring R in each of the following cases.

1. V = {a + ai | a even} and R = Z + Zi (the Gaussian integers).
2. V is the set of polynomials in Z[X] which have a zero at 0 and 3; R = Z[X].

Determine in each of the following cases if the given ideal is a prime ideal or a maximal ideal.

1. The ideal in Z[X] generated by the prime number p.
2. The ideal in the ring of Gaussian integers Z + Zi generated by the number 5.
3. The ideal ({2, X² + X + 1}) in the ring Z[X].

Let R and S be rings and let I be an ideal of R. Prove:

1. If J is an ideal in the ring S, then I × J is an ideal in the ring R × S.
2. If : R -> S is a surjective morphism, then J = (I) is an ideal of S. Also show that the induced map on the quotient rings : R/I -> S/J, (r + I) = (r) + J is a well-defined surjective morphism. Is necessarily injective?

From the Characterization of finite fields we know that the fields Z/2Z[X]/(X³ + X + 1) and Z/2Z[X]/(X³ + X² + 1) must be isomorphic. Give an explicit isomorphism.

The polynomial f = X⁴ + X + 1 generates a maximal ideal in R = Z/2Z[X] (see Section 4 in chapter 4). The quotient ring S = R/(f) is a field of order 16. Write = X + (X⁴ + X + 1).

1. Prove that X⁴ + X + 1 is irreducible. Conclude that the ideal (f) is indeed maximal. Show that each element of S can be written in a unique way as a + b + c² + d³ for certain a, b, c, d in Z/2Z.
2. Give an algorithm that decides whether two elements a₀ + a₁ + ··· + a_m^m and b₀ + b₁ + ··· + b_m^m (a₀, ..., a_m, b₀ ,..., b_n Z/2Z) are equal or not. (Instead of a₀ + a₁ + ··· + a_m^m we can also write g() with g = a₀ + a₁X + ··· + a_mX^m.)
3. Show that S = {0, , ², ..., ¹⁵ = 1}. Give the addition table for S (or S^*) in terms of powers of : Input i and j gives k with ⁱ + ^j = ^k.
4. Which of the following elements generates a subfield of S with 4 elements: ², ² + , ² + 1?
5. Write an algorithm that, with input an irreducible polynomial f in Z/pZ[X] and a generator of Z/pZ[X]/(f), returns the addition table of Z/pZ[X]/(f) in terms of powers of .

The map

1. Show that F is a morphism of rings (Q³ is the Cartesian product of three copies of the ring Q, and therefore a ring).
2. Describe the kernel of F.
3. Given rational numbers r, s and t, express the set of polynomials f in Q[X] satisfying

   f(-1) = r, f(0) = s and f(1) = t

   in terms of the kernel of F.

Consider the irreducible polynomial f = X² - 2 Z/5Z[X]. The quotient ring Z/5Z[X]/(f) is then a field with 25 elements which we denote by F. For every nonzero a in Z/5Z and for every b in Z/5Z consider the isomorphism (automorphism) F_a,b : Z/5Z[X] -> Z/5Z[X] given by F_a,b(g) = g(aX + b).

Prove that substitutions of the indicated form applied to f produce, up to a sign, every irreducible polynomial in Z/5Z[X] of degree 2 with leading coefficient 1 or -1. Use this to produce explicit isomorphisms between F and fields with 25 elements which are given as Z/5Z[X]/(h) for some irreducible polynomial h of degree 2.

Let R be a ring in which x² = x for all x in R. Prove that R is commutative. (An example of a ring R with this property is the ring of part 3 of Exercise 4.)