A field is specified by five items: 1. A set S. 2. An element of S called O. 3. An element of S...

A field is specified by five items:

1. A set S.

2. An element of S called O.

3. An element of S called 1.

4. A binary operator + defined on S that is commutative (i.e., for all a and b in S, a+b = b+a), associative (i.e., for all a, b, and c in S, a+(b+c) = (a+b)+c), for which 0 is an identity element (i.e., for all a in S, a+O = 0), and for which every element has an additive inverse with respect to 0 (i.e., for all a in S there exists a unique element b such that a+b = 0).

5. A binary operator * defined on S that is commutative, associative, for which 1 is an identity element, for which every element has a multiplicative inverse with respect to 1 (i.e., for all a≠0 in S there exists a unique element b such that a*b = 1), and which distributes over + (i.e., for all a, b, and c in S, a*(b+c) = (a*b)+(a*c) and (a+b)*c = (a*c)+(b*c).

A commutative ring is specified exactly like a field except that it is not required that every element have a multiplicative inverse. A ring is specified exactly like a commutative ring except that it is not required that multiplication be commutative.

A. Prove that the real numbers are a field.

B. Prove that the complex numbers are a field.

C. Prove that the rational numbers (all real numbers r such that there exist two integers x and y for which r = x/y) are a field.

D. Prove that the integers MOD p, for any prime number p ≥ 2, are a field.

E. Prove that the integers are a commutative ring.

F. Prove that the set of polynomials over the single variable x, where coefficients are members of a field, are a commutative ring when + adds two polynomials

G. Prove that for any n ≥ 1 the usual way (i.e., pair-wise multiplication of corresponding elements), * is standard matrix multiplication (see the first chapter), 0 is the matrix where all entries are 0, and 1 is the identity matrix (the (i,j)^{th} entry is 1 if i=j and 0 otherwise).

H. For a ring R, prove that 0 and 1 must be unique and that 0 is an annihilator; that is, for all a in R, a*O = O.