Abstract Algebra [Lecture notes] by Irena Swanson

By Irena Swanson

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A1 2 + a2 ( 2)2 + · · · + an ( 2)n : n ∈ Z≥0 , ai ∈ Z} = {a + b 2 : a, b ∈ Z}. More examples: 47 √ √ √ √ √ √ √ √ √ Z[ 2, 3] (not equal to Z[ 6]), Z[ 3, 3 3], Z[ 3, 3 3, 4 3, 5 3, . ], Z[ 21 , 13 , 41 , 15 , 61 , . ] = Q, etc. 5 Prove that the set Z[i] = {a + bi : a, b, ∈ Z} is a subring of C. The ring Z[i] is called the ring of Gaussian integers. Find, with proof, all the units in Z[i]. 6 Find all √ the units in Z[ 2]. Two of the units are 1 and 1 + 2. Let √ a + b 2 be a unit in Z[ 2], where √ a, b are integers.

It is not clear at all that it is finite. We will rewrite G with different generators, and for this we will suggestively record the relation vectors as columns in a matrix:   1 5 −3  −1 1 −3  . 1 −5 29 41 Switching the rows corresponds to switching the generators e1 , e2 , e3 of G, so the group remains unchanged. Adding an integer multiple of one row to another similarly corresponds replacing one of the generators e1 , e2 , e3 with the sum of itself plus a multiple of another. This also does not change the group.

The ring Z[i] is called the ring of Gaussian integers. Find, with proof, all the units in Z[i]. 6 Find all √ the units in Z[ 2]. Two of the units are 1 and 1 + 2. Let √ a + b 2 be a unit in Z[ 2], where √ a, b are integers. We want to find all a, b. Then √ there exist c, d ∈ Z such that (a + b 2)(c + d √2) = 1. This means that ac + 2bd = 1 √ and ad + bc = 0. Necessarily then also (a − b 2)(c − d 2) = 1, and multiplying the 2 2 2 2 four terms gives√us (a2 − 2b2 )(c √ − 2d ) = 1. We conclude that a − 2b = ±1, and in 2 is ±(a − b 2).

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