By Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted

ISBN-10: 1466567082

ISBN-13: 9781466567085

""This ebook arose from the authors' method of educating summary algebra. They position an emphasis on energetic studying and on constructing scholars' instinct via their research of examples. ... The textual content is geared up in this type of manner that it truly is attainable to start with both jewelry or groups.""

-Florentina Chirteş, *Zentralblatt MATH* 1295

summary:

""This booklet arose from the authors' method of instructing summary algebra. They position an emphasis on energetic studying and on constructing scholars' instinct via their research of examples. ... The textual content is equipped in this type of method that it really is attainable firstly both jewelry or groups.""

-Florentina Chirteş, *Zentralblatt MATH* 1295

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**Extra resources for Abstract Algebra : An Inquiry Based Approach**

**Example text**

The Division Algorithm. Let a and b be integers, with a > 0. Then there exist unique integers q and r such that b = aq + r and 0 ≤ r < a. There are a few things worth noting about the Division Algorithm before we discuss why it is true. The first is that it asserts both the existence and uniqueness of a quotient q and a remainder r, 14 Investigation 2. Divisibility of Integers but provides no actual mechanism for finding q and r. In this sense, the Division Algorithm is not an algorithm at all, and perhaps would be better called a theorem.

Although it may not be immediately apparent, 1 is different in several regards from the other numbers that meet our definition of prime. For instance, 1 has exactly one positive divisor (itself), whereas every other prime has exactly two positive divisors (1 and itself). In addition, 1·1 = 1, a property that none of the other prime numbers satisfy. ∗ The Fundamental Theorem of Arithmetic Having defined what a prime number is, we will now turn our attention to the problem of prime factorization.

3 suggests not only that gcd(a, b) exists whenever a and b are not both zero, but also that gcd(a, b) is unique—that is, there is exactly one largest integer that divides both a and b. 3. 10) that we will prove later in the investigation. 1, we were asked to calculate the greatest common divisors of several pairs of numbers. For numbers that are small enough, we can do so by simply listing the positive factors of each number and picking the largest one that appears in both lists. For instance, to find gcd(28, 42), we might first note that the positive factors of 28 are 1, 2, 4, 7, 14, and 28, and the positive factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

### Abstract Algebra : An Inquiry Based Approach by Hodge, Jonathan K.; Schlicker, Steven; Sundstrom, Ted

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