Aufgaben:Exercise 2.3Z: Polynomial Division - Revision history

Guenter at 16:17, 29 September 2022

2022-09-29T16:17:32Z

Guenter at 16:07, 29 September 2022

2022-09-29T16:07:08Z

Noah at 12:55, 31 August 2022

2022-08-31T12:55:19Z

Noah at 12:51, 31 August 2022

2022-08-31T12:51:26Z

Guenter at 16:56, 27 August 2022

2022-08-27T16:56:45Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.3Z: Polynomdivision to Aufgaben:Exercise 2.3Z: Polynomial Division

2022-08-27T15:56:15Z

Guenter moved page Aufgabe 2.3Z: Polynomdivision to Exercise 2.3Z: Polynomial Division

Javier: Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

2021-03-23T12:49:05Z

Text replacement - "Category:Aufgaben zu Kanalcodierung" to "Category:Channel Coding: Exercises"

Javier: Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

2020-07-09T12:54:30Z

Text replacement - "[[Kanalcodierung" to "[[Channel_Coding"

Guenter at 12:57, 20 May 2019

2019-05-20T12:57:03Z

Guenter at 12:50, 20 May 2019

2019-05-20T12:50:49Z

@@ Line 42: / Line 42: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The modulo 2 multiplication of the two polynomials leads to the result
+'''(1)'''&nbsp; The modulo-2 multiplication of the two polynomials leads to the result
 :$$a(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (x^3+  x+1) \cdot (x^2+1)= x^5+x^3+ x^2+ x^3+x+1 = x^5+ x^2+x+1\hspace{0.05cm}.$$
-*Thus the <u>proposed solution 2</u> is correct.
+*Thus the&nbsp; <u>proposed solution 2</u>&nbsp; is correct.
 [[File:P_ID2506__KC_Z_2_3b_neu.png|right|frame|Example 1 for polynomial division]]
 '''(2)'''&nbsp; With the abbreviations
 :$$a(x) = x^5+ x^2+x+1\hspace{0.05cm},\hspace{0.4cm}p(x) = x^3+ x+1\hspace{0.05cm},\hspace{0.4cm}q(x) = x^2+ 1$$
-and the result from subtask (1) we get $a(x) = p(x) \cdot q(x)$.
+and the result from subtask&nbsp; '''(1)'''&nbsp; we get&nbsp; $a(x) = p(x) \cdot q(x)$.
-That is: &nbsp; The divisions $a(x)/p(x)$ and $a(x)/q(x)$ are free of remainders &nbsp;
+That is: &nbsp; The divisions&nbsp; $a(x)/p(x)$&nbsp; and&nbsp; $a(x)/q(x)$&nbsp; are free of remainders &nbsp;
-&#8658; &nbsp; Correct are the <u>solutions 1 and 2</u>.
+&#8658; &nbsp; Correct are the&nbsp; <u>solutions 1 and 2</u>.
-Even without calculation one recognizes that $a(x)/x^2$ must result in a remainder. After calculation it results explicitly:
+Even without calculation one recognizes that&nbsp; $a(x)/x^2$&nbsp; must result in a remainder.&nbsp; After calculation it results explicitly:
-:$$(x^5 + x^2+x+1)/(x^2) = x^3 + 1 \hspace{0.05cm},\hspace{0.4cm}{\rm Rest}\hspace{0.15cm} r(x) = x+1\hspace{0.05cm}.$$
+:$$(x^5 + x^2+x+1)/(x^2) = x^3 + 1 \hspace{0.05cm},\hspace{0.4cm}{\rm remainder}\hspace{0.15cm} r(x) = x+1\hspace{0.05cm}.$$
-To the last proposed solution. We use for shortcut $b(x) = x^5 + x^2 + x = a(x) + 1$. This is the given quotient:
+To the last proposed solution:&nbsp; We use for shortcut&nbsp; $b(x) = x^5 + x^2 + x = a(x) + 1$.&nbsp; This is the given quotient:
 :$$b(x)/q(x) = a(x)/q(x) + 1/q(x) \hspace{0.05cm}.$$
-[[File:P_ID2505__KC_Z_2_3c.png|Right|frame|Example 2 for polynomial division]]
+[[File:P_ID2505__KC_Z_2_3c.png|Right|frame|'''Example 2'''&nbsp; for polynomial division]]
-*The first quotient $a(x)/q(x)$ gives exactly $p(x)$ without remainder, the second quotient $0$ with remainder $1$.
+*The first quotient&nbsp; $a(x)/q(x)$&nbsp; gives exactly&nbsp; $p(x)$&nbsp; without remainder,&nbsp; the second quotient&nbsp; $0$,&nbsp;  remainder&nbsp; $1$.
-*Thus the remainder of the quotient $b(x)/q(x)$ is equal to $r(x) = 1$, as the calculation in example 1 shows.
-'''(3)'''&nbsp; The polynomial division is explained in detail in example 2. Correct is the <u>proposed solution 3</u>.
+'''(3)'''&nbsp; The polynomial division is explained in detail in Example 2.&nbsp; Correct is the&nbsp; <u>proposed solution 3</u>.

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Extension_Field}}
-[[File:P_ID2504__KC_Z_2_3.png|right|frame|Multiplication and division of polynomials in&nbsp; $\rm GF(2)$]]
+[[File:P_ID2504__KC_Z_2_3.png|right|frame|Multiplication and division of polynomials in&nbsp; $\rm GF(2)$ <br><u>Note.</u> &nbsp; remainder&nbsp; $r(x)$]]
-In this exercise we deal with the multiplication and especially the division of polynomials in the Galois field&nbsp; $\rm GF(2)$. In the figure the procedure is indicated by a simple and (hopefully) self-explanatory example:
+In this exercise we deal with the multiplication and especially the division of polynomials in the Galois field&nbsp; $\rm GF(2)$.&nbsp; In the graph the procedure is indicated by a simple and&nbsp; (hopefully)&nbsp; self-explanatory example:
-* Multiplying the two polynomials&nbsp; $x^2 + 1$&nbsp; and&nbsp; $x +1$&nbsp; yields the result&nbsp; $a(x) = x^3 + x^2 + x + 1$.
+* Multiplying the two polynomials &nbsp; $x^2 + 1$ &nbsp; and &nbsp; $x +1$ &nbsp; yields the result &nbsp; $a(x) = x^3 + x^2 + x + 1$.
 * Dividing the polynomial&nbsp; $b(x) = x^3$&nbsp; by&nbsp; $p(x) = x + 1$&nbsp; gives the quotient&nbsp; $q(x) = x^2 + x$&nbsp; and the remainder&nbsp; $r(x) = x$.
 * One can check the latter result as follows:
 :$$b(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} p(x) \cdot q(x) + r(x)\hspace{0.05cm}= \big[(x+1) \cdot (x^2+x)\big] +x =\big[x^3+ x^2+x^2+ x\big] +x = x^3\hspace{0.05cm}.$$
-Hint:
+Hint:&nbsp; This exercise belongs to the chapter&nbsp; [[Channel_Coding/Extension_Field| "Extension field"]].
@@ Line 25: / Line 26: @@
 - $a(x) = x^6 + x^3 + x^2 + 1$-
-{Which of the polynomial divisions do not yield a remainder&nbsp; $r(x)$?
+{Which of the polynomial divisions do not yield a remainder&nbsp; $r(x) \ne 0$?
 |type="[]"}
 + $(x^5 + x^2 + x + 1)/(x^3 + x + 1)$.
@@ Line 32: / Line 33: @@
 - $(x^5 + x^2 + x)/(x^2 + 1)$.
-{It is&nbsp;  $a(x) = x^6 + x^5 + 1$&nbsp;  and&nbsp;  $p(x) = x^3 + x^2 + 1$. <br>Determine&nbsp; $q(x)$&nbsp; and&nbsp; $r(x)$&nbsp; according to the description equation&nbsp; $a(x) = p(x) \cdot q(x) + r(x)$.
+{It is &nbsp;  $a(x) = x^6 + x^5 + 1$ &nbsp;  and &nbsp;  $p(x) = x^3 + x^2 + 1$. <br>Determine&nbsp; $q(x)$&nbsp; and&nbsp; $r(x)$&nbsp; according to the description equation &nbsp; $a(x) = p(x) \cdot q(x) + r(x)$.
 |type="()"}
 - $q(x) = x^3 + x^2 + 1, \hspace{0.2cm} r(x) = 0$,

@@ Line 17: / Line 17: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Welches Ergebnis liefert&nbsp;  $a(x) = (x^3 + x + 1) \cdot (x^2 + 1)$?
+{What is the result&nbsp; $a(x) = (x^3 + x + 1) \cdot (x^2 + 1)$?
 |type="()"}
 - $a(x) = x^5 + x^3 + x^2 + 1$,
@@ Line 25: / Line 25: @@
 - $a(x) = x^6 + x^3 + x^2 + 1$-
-{Welche der Polynomdivisionen ergeben keinen Rest&nbsp;  $r(x)$?
+{Which of the polynomial divisions do not yield a remainder&nbsp; $r(x)$?
 |type="[]"}
 + $(x^5 + x^2 + x + 1)/(x^3 + x + 1)$.
@@ Line 32: / Line 32: @@
 - $(x^5 + x^2 + x)/(x^2 + 1)$.
-{Es sei&nbsp;  $a(x) = x^6 + x^5 + 1$&nbsp;  und&nbsp;  $p(x) = x^3 + x^2 + 1$. <br>Bestimmen Sie&nbsp;  $q(x)$&nbsp;  und&nbsp;  $r(x)$&nbsp;  entsprechend der Beschreibungsgleichung&nbsp;  $a(x) = p(x) \cdot q(x) + r(x)$.
+{It is&nbsp;  $a(x) = x^6 + x^5 + 1$&nbsp;  and&nbsp;  $p(x) = x^3 + x^2 + 1$. <br>Determine&nbsp; $q(x)$&nbsp; and&nbsp; $r(x)$&nbsp; according to the description equation&nbsp; $a(x) = p(x) \cdot q(x) + r(x)$.
 |type="()"}
 - $q(x) = x^3 + x^2 + 1, \hspace{0.2cm} r(x) = 0$,
@@ Line 39: / Line 39: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Die Modulo&ndash;2&ndash;Multiplikation der beiden Polynome führt zum Ergebnis
+'''(1)'''&nbsp; The modulo 2 multiplication of the two polynomials leads to the result
 :$$a(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} (x^3+  x+1) \cdot (x^2+1)= x^5+x^3+ x^2+ x^3+x+1 = x^5+ x^2+x+1\hspace{0.05cm}.$$
-*Richtig ist somit der <u>Lösungsvorschlag 2</u>.
+*Thus the <u>proposed solution 2</u> is correct.
-*Der letzte Lösungsvorschlag kann schon alleine deshalb nicht simmen, da der Grad des Produktpolynoms ungleich $5$ wäre.
+*The last solution suggestion cannot simmen already alone, since the degree of the product polynomial would be unequal $5$.
-[[File:P_ID2506__KC_Z_2_3b_neu.png|right|frame|Beispiel 1 zur Polynomdivision]]
+[[File:P_ID2506__KC_Z_2_3b_neu.png|right|frame|Example 1 for polynomial division]]
-'''(2)'''&nbsp; Mit den Abkürzungen
+'''(2)'''&nbsp; With the abbreviations
 :$$a(x) = x^5+ x^2+x+1\hspace{0.05cm},\hspace{0.4cm}p(x) = x^3+ x+1\hspace{0.05cm},\hspace{0.4cm}q(x) = x^2+ 1$$
-und dem Ergebnis aus der Teilaufgabe (1) erhält man $a(x) = p(x) \cdot q(x)$.
+and the result from subtask (1) we get $a(x) = p(x) \cdot q(x)$.
-Das heißt: &nbsp; Die Divisionen $a(x)/p(x)$ und $a(x)/q(x)$ sind restfrei möglich &nbsp;
+That is: &nbsp; The divisions $a(x)/p(x)$ and $a(x)/q(x)$ are free of remainders &nbsp;
-&#8658; &nbsp; Richtig sind die <u>Lösungsvorschläge 1 und 2</u>.
+&#8658; &nbsp; Correct are the <u>solutions 1 and 2</u>.
-Auch ohne Rechnung erkennt man, dass $a(x)/x^2$ einen Rest ergeben muss. Nach Rechnung ergibt sich explizit:
+Even without calculation one recognizes that $a(x)/x^2$ must result in a remainder. After calculation it results explicitly:
 :$$(x^5 + x^2+x+1)/(x^2) = x^3 + 1 \hspace{0.05cm},\hspace{0.4cm}{\rm Rest}\hspace{0.15cm} r(x) = x+1\hspace{0.05cm}.$$
-Zum letzten Lösungsvorschlag. Wir verwenden zur Abkürzung $b(x) = x^5 + x^2 + x = a(x) + 1$. Damit ist der vorgegebene Quotient:
+To the last proposed solution. We use for shortcut $b(x) = x^5 + x^2 + x = a(x) + 1$. This is the given quotient:
 :$$b(x)/q(x) = a(x)/q(x) + 1/q(x) \hspace{0.05cm}.$$
-[[File:P_ID2505__KC_Z_2_3c.png|Right|frame|Beispiel 2 zur Polynomdivision]]
+[[File:P_ID2505__KC_Z_2_3c.png|Right|frame|Example 2 for polynomial division]]
-*Der erste Quotient $a(x)/q(x)$ ergibt entsprechend der Teilaufgabe (2) genau $p(x)$ ohne Rest, der zweite Quotient $0$ mit Rest $1$.
+*The first quotient $a(x)/q(x)$ gives exactly $p(x)$ without remainder, the second quotient $0$ with remainder $1$.
-*Somit ist hier der Rest des Quotienten $b(x)/q(x)$ gleich $r(x) = 1$, wie auch die Rechnung im Beispiel 1 zeigt.
+*Thus the remainder of the quotient $b(x)/q(x)$ is equal to $r(x) = 1$, as the calculation in example 1 shows.
-'''(3)'''&nbsp; Die Polynomdivision ist im Beispiel 2 ausführlich erläutert. Richtig ist der <u>Lösungsvorschlag 3</u>.
+'''(3)'''&nbsp; The polynomial division is explained in detail in example 2. Correct is the <u>proposed solution 3</u>.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Kanalcodierung/Erweiterungskörper}}
+{{quiz-Header|Buchseite=Channel_Coding/Extension_Field}}
-[[File:P_ID2504__KC_Z_2_3.png|right|frame|Multiplikation und Division von Polynomen  in&nbsp; $\rm GF(2)$]]
+[[File:P_ID2504__KC_Z_2_3.png|right|frame|Multiplication and division of polynomials in&nbsp; $\rm GF(2)$]]
-In dieser Aufgabe beschäftigen wir uns mit der Multiplikation und insbesondere der Division von Polynomen im Galoisfeld&nbsp;  $\rm GF(2)$. In der Abbildung ist jeweils die Vorgehensweise an einem einfachen und (hoffentlich) selbsterklärenden Beispiel angedeutet:
+In this exercise we deal with the multiplication and especially the division of polynomials in the Galois field&nbsp; $\rm GF(2)$. In the figure the procedure is indicated by a simple and (hopefully) self-explanatory example:
-* Die Multiplikation der beiden Polynome&nbsp;  $x^2 + 1$&nbsp;  und&nbsp;  $x +1$&nbsp;  liefert das Ergebnis&nbsp;  $a(x) = x^3 + x^2 + x + 1$.
+* Multiplying the two polynomials&nbsp; $x^2 + 1$&nbsp; and&nbsp; $x +1$&nbsp; yields the result&nbsp; $a(x) = x^3 + x^2 + x + 1$.
-* Die Division des Polynoms&nbsp;  $b(x) = x^3$&nbsp;  durch&nbsp;  $p(x) = x + 1$&nbsp;  liefert den Quotienten&nbsp;  $q(x) = x^2 + x$&nbsp;  und den Rest&nbsp;  $r(x) = x$.
+* Dividing the polynomial&nbsp; $b(x) = x^3$&nbsp; by&nbsp; $p(x) = x + 1$&nbsp; gives the quotient&nbsp; $q(x) = x^2 + x$&nbsp; and the remainder&nbsp; $r(x) = x$.
-* Man kann das letztere Ergebnis wie folgt überprüfen:
+* One can check the latter result as follows:
 :$$b(x) \hspace{-0.15cm} \ = \ \hspace{-0.15cm} p(x) \cdot q(x) + r(x)\hspace{0.05cm}= \big[(x+1) \cdot (x^2+x)\big] +x =\big[x^3+ x^2+x^2+ x\big] +x = x^3\hspace{0.05cm}.$$
-''Hinweis:''
+Hint:
-* Die Aufgabe gehört zum Kapitel&nbsp;  [[Channel_Coding/Erweiterungsk%C3%B6rper| Erweiterungskörper]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Extension_Field| "Extension field"]].

← Older revision		Revision as of 16:56, 27 August 2022
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−	[[Category:Channel Coding: Exercises\|^2.2 ~~Erweiterungskörper~~^]]	+	[[Category:Channel Coding: Exercises\|^2.2 Extension Field^]]

← Older revision	Revision as of 15:56, 27 August 2022
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← Older revision		Revision as of 12:49, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^2.2 Erweiterungskörper^]]	+	[[Category:Channel Coding: Exercises\|^2.2 Erweiterungskörper^]]

← Older revision		Revision as of 12:54, 9 July 2020
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	''Hinweis:''		''Hinweis:''
−	* Die Aufgabe gehört zum Kapitel  [[~~Kanalcodierung~~/Erweiterungsk%C3%B6rper\| Erweiterungskörper]].	+	* Die Aufgabe gehört zum Kapitel  [[Channel_Coding/Erweiterungsk%C3%B6rper\| Erweiterungskörper]].