Math 515 homework #2 Due date: 19980123 D. J. Bernstein 19980116 Let R be a ring under 0, 1, −, +, ·. A subset S ⊆ R is a subring of R if S is closed under 0, 1, −, +, ·. In other words, 0 ∈ S; 1 ∈ S; −a ∈ S whenever a ∈ S; a + b ∈ S whenever a ∈ S and b ∈ S; and ab ∈ S whenever a ∈ S and b ∈ S. Then S is a ring under 0, 1, − on S, + on S × S, · on S × S. For example, Q is a subring of C. Exercise 3. Let R be a ring. Let E be a subset of R. Let S be the intersection of all subrings of R that contain E. Prove that S is a subring of R. (Hence S is the smallest subring of R containing E.)

Fields. An element u of a ring R is a unit in R if uR = R. Here uR means {ur : r ∈ R}. In other words, u has a multiplicative inverse. This inverse is unique; it is written u^{−1}. The set of units of R is written R^{∗}. R^{∗} is a group under 1, (u ↦ u^{−1}), ·. For example, Z^{∗} = {−1, 1}. R is a field if 0 ≠ 1 and R^{∗} = R − {0}. For example, Q and R and Z/2 are fields; Z is not. A subfield of R means a subring of R that is a field under R’s operations. Exercise 4. Let R be a field. Let E be a subset of R. Let S be the intersection of all subfields of R that contain E. Prove that S is a subfield of R. (Hence S is the smallest subfield of R containing E.) Exercise 5. Give an example of a field R and a subset E for which the smallest subfield of R containing E is not the smallest subring of R containing E.