By Connell E H

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Exercise Let C = {a + bi : a, b ∈ R}. Since R is a subring of C, there exists a homomorphism h : R[x] → C which sends x to i, and this h is surjective. Show ker(h) = (x2 + 1)R[x ] and thus R[x ]/(x 2 + 1) ≈ C. , to obtain C, adjoin x to R and set x2 = −1. Exercise Z2 [x ]/(x 2 + x + 1) has 4 elements. Write out the multiplication table for this ring and show that it is a field. Exercise Show that, if R is a domain, the units of R[x ] are just the units of R. Thus if F is a field, the units of F [x ] are the non-zero constants.

Am,n A matrix may be viewed as m n-dimensional row vectors or as n m-dimensional column vectors. A matrix is said to be square if it has the same number of rows as columns. Square matrices are so important that they have a special notation, Rn = Rn,n . Rn is defined to be the additive abelian group R × R × · · · × R. To emphasize that Rn does not have a ring structure, we use the “sum” notation, Rn = R ⊕ R ⊕ · · · ⊕ R. , to identify Rn with Rn,1 . If the elements of Rn are written as row vectors, Rn is identified with R1,n .

What theorems in calculus show that H and K are subgroups of G? What theorem shows that K is a subset (and thus subgroup) of H? Order Suppose G is a multiplicative group. If G has an infinite number of Chapter 2 Groups 23 elements, we say that o(G), the order of G, is infinite. If G has n elements, then o(G) = n. Suppose a ∈ G and H = {ai : i ∈ Z}. H is an abelian subgroup of G called the subgroup generated by a. , the order of the subgroup generated by a. Let f : Z → H be the surjective function defined by f (m) = am .