Let R be a commutative ring with unity and let U(R) denote the setof units of R. Prove that U(R) is a group under the multiplication ofR. (This group is called the group of units of R.)

If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in R2. Provide a origin is a subgroup. Now, if BE R2, then 8+ L represents the left coset geometric description of this coset and explain how you arrived at this description.

Find the group of automorphisms (Galois group) of the following fields E: (a) Q(√3, √5) (b) Q(w), where w is a primitive cube root of unity (c) Q(w), where w is a primitive fifth root of unity

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