 
Summary: Math 311001 201110
Assignment # 1 (due: Jan 27th)
1. Assume the existence of N with its addition and multiplication.
(a) Complete the details for the construction of Z (i.e. show that there
exists a commutative ring whose elements are exactly the natural
numbers, their additive inverses, and zero; show that addition and
multiplication extend to Z and they satisfy axioms IVI (with the
exception of multiplicative inverses).
(b) Complete the details for the construction of Q (i.e. show that
there exists a field satisfying IVI, that contains Z and such that
every element is an integer multiple of the multiplicative inverse
of an integer).
(c) Complete the details for the construction of R (i.e. show that
there exists a field satisfying IVII) and prove that such a field is
unique (up to isomorphism of fields, i.e. if there is another field R
satisfying IVII, then there is a bijection R R that preserves
sums and products).
2. Consider the space R × R, and the function
d((x, y), (z, w)) =
x  z + y + w if x = z
