Abstract Algebra: Preliminaries - Revision history

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Licensing

Lila: /* Algebra on \Z_m */

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Algebra on \Z_m

Lila: /* Algebra on \Z_m */

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Algebra on \Z_m

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2021-11-19T17:59:25Z

Algebra on \Z_m

Lila: /* Algebra on \Z_m */

2021-11-19T17:57:05Z

Algebra on \Z_m

Lila: /* Algebra on \Z_m */

2021-11-19T17:55:05Z

Algebra on \Z_m

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2021-11-19T17:52:17Z

Algebra on \Z_m

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2021-11-19T17:40:36Z

Algebra on \Z_m

Lila: /* Fundamental Theorem of Arithmetic */

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Fundamental Theorem of Arithmetic

Lila: /* Fundamental Theorem of Arithmetic */

2021-11-19T17:24:14Z

Fundamental Theorem of Arithmetic

← Older revision		Revision as of 18:38, 19 November 2021
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	* [https://en.wikipedia.org/wiki/Prime_number Prime number, Wikipedia] under a CC BY-SA license		* [https://en.wikipedia.org/wiki/Prime_number Prime number, Wikipedia] under a CC BY-SA license
	* [http://mathonline.wikidot.com/abstract-algebra Abstract Algebra, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/abstract-algebra Abstract Algebra, mathonline.wikidot.com] under a CC BY-SA license
		+	* [https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/7%3A_Equivalence_Relations/7.4%3A_Modular_Arithmetic Modular Arithmetic, LibreTexts: Mathematics] under a CC BY-SA license

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 </blockquote>
-<p>Because of Corollary 7.19, we know that the following formal definition of addition and multiplication of congruence classes in <math>\Z_n</math> is independent of the choice of the elements we choose from each class. We say that these definitions of addition and multiplication are <strong>well defined</strong>. </p>
+<p>Now, we know that the following formal definition of addition and multiplication of congruence classes in <math>\Z_n</math> is independent of the choice of the elements we choose from each class. We say that these definitions of addition and multiplication are <strong>well defined</strong>. </p>
 ==GCD, LCM, and Bézout's identity==

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 <table class="mt-responsive-table">
          <tr>
              <td class="mt-noheading lt-math-7078">
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              </td>
          </tr>
 </table>

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 <p>These are illustrations of general properties that we have already proved in Theorem 3.28. We repeat the statement of the theorem here because it is so important for defining the operations of addition and multiplication in <math>\Z_n</math>.</p>
-<div class="box-definition"><div mt-section-origin="Bookshelves/Mathematical_Logic_and_Proof/Book:_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/7:_Equivalence_Relations/7.4:_Modular_Arithmetic" class="mt-section"><span id="Theorem_3.28"></span><h5 class="box-legend lt-math-7078 editable"><span class="lt-icon-default">Theorem 3.28</span></h5>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 <p>Let <math>n</math> be a natural number and let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be integers. Then</p>
@@ Line 348: / Line 348: @@
 </ol>
-<dl>
+</blockquote>
 <p>Since <math>x \equiv y</math> (mod <math>n</math>) if and only if <math>[x] = [y]</math>, we can restate the result of this Theorem 3.28 in terms of congruence classes in <math>\Z_n</math>.</p>
-<div class="box-definition"><div mt-section-origin="Bookshelves/Mathematical_Logic_and_Proof/Book:_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/7:_Equivalence_Relations/7.4:_Modular_Arithmetic" class="mt-section"><span id="Corollary_7.19."></span><h5 class="box-legend lt-math-7078 editable"><span class="lt-icon-default">Corollary 7.19.</span></h5>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 <p>Let <math>n</math> be a natural number and let <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> be integers. Then, in <math>\Z_n</math>.</p>
@@ Line 367: / Line 361: @@
      <li class="lt-math-7078">If <math>[a] = [b]</math> and <math>m \in \N</math>, then <math>[a]^m = [b]^m</math>.</li>
 </ol>
-</div></div>
+</blockquote>
 <p>Because of Corollary 7.19, we know that the following formal definition of addition and multiplication of congruence classes in <math>\Z_n</math> is independent of the choice of the elements we choose from each class. We say that these definitions of addition and multiplication are <strong>well defined</strong>. </p>

@@ Line 300: / Line 300: @@
 <p>One of the basic problems dealt with in modern algebra is to determine if the arithmetic operations on one set &ldquo;transfer&rdquo; to a related set. In this case, the related set is <math>\Z_n</math>. For example, in the integers modulo 5, <math>\Z_5</math>, is it possible to add the congruence classes [4] and [2] as follows?</p>
-<p><math>\begin{array} {rcl} {[4] \oplus [2]} &amp;= &amp; {[4 + 2]} \\ {} &amp;= &amp; {[6]} \\ {} &amp;= &amp; {[1].} \end{array}</math></p>
+<p><math>\begin{array} {rcl} {[4] \oplus [2]} & = & {[4 + 2]} \\ {} & = & {[6]} \\ {} & = & {[1].} \end{array}</math></p>
 <p>We have used the symbol &#778; to denote addition in <math>\Z_5</math> so that we do not confuse it with addition in <math>\Z</math>. This looks simple enough, but there is a problem. The congruence classes [4] and [2] are not numbers, they are infinite sets. We have to make sure that we get the same answer no matter what element of [4] we use and no matter what element of [2] we use. For example,</p>
@@ Line 309: / Line 309: @@
 <p>Do we get the same result if we add [9] and [7] in the way we did when we added [4] and [2]? The following computation confirms that we do:</p>
-<p><math>\begin{array} {rcl} {[9] \oplus [7]} &amp;= &amp; {[9 + 7]} \\ {} &amp;= &amp; {[16]} \\ {} &amp;= &amp; {[1].} \end{array}</math></p>
+<p><math>\begin{array} {rcl} {[9] \oplus [7]} & = & {[9 + 7]} \\ {} & = & {[16]} \\ {} & = & {[1].} \end{array}</math></p>
 <p>The left side shows the properties in terms of the congruence relation and the right side shows the properties in terms of the congruence classes.</p>

← Older revision		Revision as of 17:40, 19 November 2021
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	==Algebra on <math>\Z_m</math>==		==Algebra on <math>\Z_m</math>==
		+	<p>We will now look at the additive group of integers modulo <span class="math-inline"><math>n</math></span> denoted <span class="math-inline"><math>\mathbb{Z}/n\mathbb{Z} = \{ 0, 1, 2, ..., n - 1 \}</math></span> for integers <span class="math-inline"><math>n > 1</math></span>. For each of these groups, define the operation of <span class="math-inline"><math>+</math></span> for <span class="math-inline"><math>x, y \in \mathbb{Z}/n\mathbb{Z}</math></span> by <span class="math-inline"><math>x + y</math></span> to equal <span class="math-inline"><math>x + y \pmod n</math></span> (where the righthand <span class="math-inline"><math>+</math></span> denotes usual addition).</p>
		+	<p>The operation <span class="math-inline"><math>+</math></span> above has a very nice interpretation. For any <span class="math-inline"><math>n > 1</math></span> we can interpret the operation of <span class="math-inline"><math>+</math></span> on <span class="math-inline"><math>\mathbb{Z}/n\mathbb{Z}</math></span> for <span class="math-inline"><math>x + y</math></span> by considering a circle numbered <span class="math-inline"><math>0</math></span> through <span class="math-inline"><math>n - 1</math></span> clockwise, starting at <span class="math-inline"><math>x</math></span>, and then traveling clockwise around the circle a distance of <span class="math-inline"><math>y</math></span> to get <span class="math-inline"><math>x + y</math></span> as illustrated in the following diagram:</p>
		+	<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/the-additive-group-of-integers-modulo-n-z-nz/Screen%20Shot%202015-09-05%20at%2011.03.05%20AM.png" alt="Screen%20Shot%202015-09-05%20at%2011.03.05%20AM.png" class="image" /></div>
		+	<p>We will now show that <span class="math-inline"><math>(\mathbb{Z}/n\mathbb{Z}, +)</math></span> is a group. For <span class="math-inline"><math>x, y \in \mathbb{Z}/n\mathbb{Z}</math></span> we have the <span class="math-inline"><math>x + y</math></span> is the remainder of the sum <span class="math-inline"><math>x + y</math></span> on division by <span class="math-inline"><math>n</math></span>. The only possible remainders are <span class="math-inline"><math>0, 1, ..., n - 1</math></span> and so <span class="math-inline"><math>(x + y) \in \mathbb{Z}/n\mathbb{Z}</math></span>, so <span class="math-inline"><math>\mathbb{Z}/n\mathbb{Z}</math></span> is closed under <span class="math-inline"><math>+</math></span>.</p>
		+	<p>It's not hard to intuitively see that the remainder of <span class="math-inline"><math>x + (y + z)</math></span> upon division by <span class="math-inline"><math>n</math></span> is equal to the remainder of <span class="math-inline"><math>(x + y) + z</math></span> for all <span class="math-inline"><math>x, y, z \in \mathbb{Z}/n\mathbb{Z}</math></span>, so indeed, <span class="math-inline"><math>x + (y + z) = (x + y) + z</math></span>, so <span class="math-inline"><math>+</math></span> is associative.</p>
		+	<p>For each <span class="math-inline"><math>x \in \mathbb{Z}/n\mathbb{Z}</math></span> we note that the remainder of <span class="math-inline"><math>x + 0</math></span> is equal to the remainder of <span class="math-inline"><math>x</math></span> upon division by <span class="math-inline"><math>n</math></span> and the remainder of <span class="math-inline"><math>0 + x</math></span> is equal to the remainder of <span class="math-inline"><math>x</math></span> upon division by <span class="math-inline"><math>n</math></span>. In other words, <span class="math-inline"><math>x + 0 = x</math></span> and <span class="math-inline"><math>0 + x = x</math></span>. Furthermore, <span class="math-inline"><math>0 \in \mathbb{Z}/n\mathbb{Z}</math></span>, so <span class="math-inline"><math>0</math></span> is the identity element.</p>
		+	<p>Lastly, if <span class="math-inline"><math>x \in \mathbb{Z}/n\mathbb{Z}</math></span> then <span class="math-inline"><math>-x \in \mathbb{Z}/n\mathbb{Z}</math></span> is such that <span class="math-inline"><math>x + (-x) = 0</math></span> and <span class="math-inline"><math>(-x) + x = 0</math></span>.</p>
		+	<p>Thus we conclude that <span class="math-inline"><math>(\mathbb{Z}/n\mathbb{Z}, +)</math></span> is a group.</p>

	==GCD, LCM, and Bézout's identity==		==GCD, LCM, and Bézout's identity==

@@ Line 405: / Line 405: @@
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
-<td><strong>Theorem 1 (The Unique Prime Factorization Theorem):</strong> If <span class="math-inline"><math>n \in \mathbb{Z}</math></span> and <span class="math-inline"><math>n > 1</math></span> then <span class="math-inline"><math>n</math></span> can be written as a product of primes uniquely apart from the ordering of the primes in the product; that is, <math> n = p_1p_2...p_k </math> for unique primes <math> p_i </math>.</td>
+<td><strong>Theorem 1 (The Unique Prime Factorization Theorem):</strong> If <span class="math-inline"><math>n \in \mathbb{Z}</math></span> and <span class="math-inline"><math>n > 1</math></span> then <span class="math-inline"><math>n</math></span> can be written as a product of primes uniquely apart from the ordering of the primes in the product (<math> n = p_1p_2...p_k </math> for unique primes <math> p_1, p_2, ..., p_k </math>).</td>
 </blockquote>
 <p><em>If <span class="math-inline"><math>n \in \mathbb{Z}</math></span> and <span class="math-inline"><math>n < -1</math></span> then <span class="math-inline"><math>n</math></span> can also be written as a product of primes multiplied by <span class="math-inline"><math>-1</math></span>.</em></p>