Aufgaben:Exercise 2.1: Group, Ring, Field - Revision history

Guenter at 13:17, 27 August 2022

2022-08-27T13:17:54Z

Guenter at 12:37, 27 August 2022

2022-08-27T12:37:11Z

Guenter at 14:41, 24 August 2022

2022-08-24T14:41:43Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.1: Gruppe, Ring, Körper to Aufgaben:Exercise 2.1: Group, Ring, Field

2022-08-24T14:36:39Z

Guenter moved page Aufgabe 2.1: Gruppe, Ring, Körper to Exercise 2.1: Group, Ring, Field

Noah at 13:24, 24 August 2022

2022-08-24T13:24:49Z

Noah at 13:24, 24 August 2022

2022-08-24T13:24:15Z

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Guenter at 12:45, 22 August 2022

2022-08-22T12:45:52Z

Javier: Text replacement - "„" to """

2021-05-28T14:27:10Z

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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"

@@ Line 77: / Line 77: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With the number set $Z_3 = \{0, \, 1, \, 2\}$ describes
+'''(1)'''&nbsp; With the number set&nbsp; $Z_3 = \{0, \, 1, \, 2\}$&nbsp; describes
-* the table A3 the additive group $(Z_3, \, +)$,
+* the table A3 the additive group&nbsp; $(Z_3, \, +)$,
-* the table M3 the multiplicative group $(Z_3, \, \cdot)$.
+* the table M3 the multiplicative group&nbsp; $(Z_3, \, \cdot)$.
-&#8658; &nbsp; <u>Solution suggestion 1 and 2</u>. The solution suggestions 3 and 4 do not apply here, because only one arithmetic operation (addition or multiplication) is defined for each group.
+&#8658; &nbsp; <u>Solution suggestion 1 and 2</u>.&nbsp;
 '''(4)'''&nbsp; The zero element is never a primitive element. Also $z_1 = 1$ is not a primitive element, because then $q = 3$ would have to hold:
 :$$z_1^1 \hspace{0.15cm}{\rm mod\hspace{0.15cm}} 3\hspace{0.1cm} \ne 1  \hspace{0.05cm},\hspace{0.3cm}z_1^2 \hspace{0.15cm}{\rm mod\hspace{0.15cm}} 3\hspace{0.1cm} = 1
 \hspace{0.05cm}.$$
-On the other hand, $z_2 = 2$ is a primitive element because of
+On the other hand,&nbsp; $z_2 = 2$&nbsp; is a primitive element because of
 :$$2^1 \hspace{0.15cm}{\rm mod\hspace{0.15cm}} 3\hspace{0.1cm} = 2  \hspace{0.05cm},\hspace{0.3cm}2^2 \hspace{0.15cm}{\rm mod\hspace{0.15cm}} 3\hspace{0.1cm} = 1
 \hspace{0.05cm}.$$
-The correct solution is <u>solution suggestion 3</u>.
+The correct solution is&nbsp; <u>solution suggestion 3</u>.
-'''(5)'''&nbsp; The set $\{0, \, 1, \, 2, \, 3\}$ has <u>no primitive element</u> and accordingly does not satisfy the requirements of a Galois field:
+'''(5)'''&nbsp; The set&nbsp; $\{0, \, 1, \, 2, \, 3\}$&nbsp; has <u>no primitive element</u> and accordingly does not satisfy the requirements of a Galois field:
 :$$z_1 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1\text{:}\hspace{0.35cm}  1^1 = 1\hspace{0.05cm},\hspace{0.1cm}1^2 = 1\hspace{0.05cm},\hspace{0.1cm}1^3 = 1\hspace{0.05cm},$$
 :$$z_2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 2\text{:}\hspace{0.35cm}  2^1 = 2\hspace{0.05cm},\hspace{0.1cm}2^2 \hspace{0.15cm}{\rm mod}\hspace{0.15cm}4 = 0 \hspace{0.05cm},\hspace{0.1cm}2^3 \hspace{0.15cm}{\rm mod}\hspace{0.15cm}4 = 0\hspace{0.05cm},$$

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Some_Basics_of_Algebra}}
-[[File:EN_KC_A_2_1.png|right|frame|"Addition" and "Multiplication" for&nbsp; $q = 3$&nbsp; and&nbsp; $q = 4$.]]
+[[File:EN_KC_A_2_1.png|right|frame|"Addition" and "Multiplication" <br> &nbsp; &nbsp; for&nbsp; $q = 3$&nbsp; and&nbsp; $q = 4$.]]
-In the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra|"theory section"]]&nbsp; to this chapter, various algebraic terms were defined. For what follows, we assume that all sets consist of&nbsp; $q$&nbsp; elements each, where here either&nbsp; $q = 3$&nbsp; or&nbsp; $q = 4$&nbsp; shall hold. Then holds:
+In the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra|"theory section"]]&nbsp; to this chapter,&nbsp; various algebraic terms were defined.&nbsp; For what follows,&nbsp; we assume that all sets consist of&nbsp; $q$&nbsp; elements each,&nbsp; where here either&nbsp; $q = 3$&nbsp; or&nbsp; $q = 4$.&nbsp; Then holds:
-* A&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_group|algebraic group]]&nbsp; is a finite set&nbsp; $G = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; together with a linking rule defined between all elements. An additive group is denoted by&nbsp; $(G, \ +)$&nbsp; a multiplicative one by&nbsp; $(G, \ \cdot)$.
+* An&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_group|"algebraic group"]]&nbsp; is a finite set&nbsp; $G = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; together with a linking rule defined between all elements.&nbsp; An&nbsp; "additive group"&nbsp; is denoted by&nbsp; $(G, \ +)$,&nbsp; a multiplicative one by&nbsp; $(G, \ \cdot)$.
-* A&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_ring|"algebraic ring"]]&nbsp; denotes a set&nbsp; $R = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; together with two arithmetic operations defined therein, namely addition&nbsp; ("$+$")&nbsp; and multiplication&nbsp; ("$\hspace{0.05cm}\cdot\hspace{0.05cm}$").
+* An&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_and_examples_of_an_algebraic_ring|"algebraic ring"]]&nbsp; denotes a set&nbsp; $R = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; together with two arithmetic operations defined therein,&nbsp; namely addition&nbsp; ("$+$")&nbsp; and multiplication&nbsp; ("$\hspace{0.05cm}\cdot\hspace{0.05cm}$").
-* A&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Group.2C_ring.2C_field_-_basic_algebraic_concepts|"algebraic field"]]&nbsp; is a ring where division is additionally allowed and the commutative law is satisfied.
+* An&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Group.2C_ring.2C_field_-_basic_algebraic_concepts|"algebraic field"]]&nbsp; is a ring where division is additionally allowed and the commutative law is satisfied.
-Since we consider here finite sets only, a field is at the same time a Galois field&nbsp; ${\rm GF}(q)$&nbsp; of order&nbsp; $q$.
+Since we consider here finite sets only,&nbsp; a&nbsp; "field"&nbsp; is at the same time a&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field|"Galois field"]]&nbsp; ${\rm GF}(q)$&nbsp; of order&nbsp; $q$.
-An essential property of the Galois field&nbsp; ${\rm GF}(q) = \{\hspace{0.1cm}z_0,\hspace{0.1cm} z_1,\hspace{0.05cm}\text{...} \hspace{0.1cm} , \hspace{0.1cm}z_{q-1}\}$&nbsp; is that it has at least one primitive element. An element&nbsp; $z_i &ne; 0$&nbsp; is said to be primitive if the following condition is satisfied &nbsp;$(k$ is integer$)$:
+An essential property of the Galois field&nbsp; ${\rm GF}(q) = \{\hspace{0.1cm}z_0,\hspace{0.1cm} z_1,\hspace{0.05cm}\text{...} \hspace{0.1cm} , \hspace{0.1cm}z_{q-1}\}$&nbsp; is that it has at least one primitive element:&nbsp; An element&nbsp; $z_i &ne; 0$&nbsp; is said to be&nbsp; "primitive"&nbsp; if the following condition is satisfied &nbsp;$(k$&nbsp; is an integer$)$:
 :$$z_i^k  \hspace{0.15cm}{\rm mod}\hspace{0.15cm}q = \left\{ \begin{array}{c} \ne 1\\
    \end{array} \right.\quad
-\begin{array}{*{1}c} {\rm f\ddot{u}r} \hspace{0.15cm}1 \le k < q-1
+\begin{array}{*{1}c} {\rm for} \hspace{0.15cm}1 \le k < q-1
-\\  {\rm f\ddot{u}r} \hspace{0.15cm}k = q-1 \\ \end{array}
+\\  {\rm for} \hspace{0.15cm}k = q-1 \\ \end{array}
-\hspace{0.35cm} \Rightarrow \hspace{0.35cm} z_i \hspace{0.15cm}{\rm ist \hspace{0.15cm}ein\hspace{0.15cm} primitives \hspace{0.15cm}Element}
+\hspace{0.35cm} \Rightarrow \hspace{0.35cm} z_i \hspace{0.15cm}{\rm is \hspace{0.15cm}a\hspace{0.15cm} primitive \hspace{0.15cm}element}
 \hspace{0.05cm}. $$
-Only for a primitive element&nbsp; $z_i$&nbsp; the arithmetic oparation&nbsp; $z_i^k$&nbsp; $($with&nbsp; $k = 1, \, 2, \, 3, \, \hspace{0.05cm}\text{...} )$ yields all elements of the Galois field except the zero element $z_0 = 0$.
+Only for a primitive element&nbsp; $z_i$&nbsp;
+*the arithmetic oparation&nbsp; $z_i^k$&nbsp; $($with&nbsp; $k = 1, \, 2, \, 3, \, \hspace{0.05cm}\text{...} )$&nbsp; yields all elements of the Galois field
+*except the zero element $z_0 = 0$.
@@ Line 28: / Line 28: @@
-Hints:
+* Note that for group, ring and field with each&nbsp; $q$&nbsp; elements,&nbsp; the arithmetic operations&nbsp; "$+$"&nbsp; and&nbsp; "$\hspace{0.05cm}\cdot\hspace{0.05cm}$"&nbsp; are to be understood modulo&nbsp; $q$.
@@ Line 39: / Line 39: @@
 ===Questions===
 <quiz display=simple>
-{Which of the given tables describe a group?
+{Which of the given tables describe a&nbsp; '''group'''?
 |type="[]"}
 + Table &nbsp;$\rm A3$,
@@ Line 46: / Line 46: @@
 - Table &nbsp;$\rm A4$&nbsp; and table &nbsp;$\rm M4$&nbsp; together.
-{Which of the given tables describe a ring?
+{Which of the given tables describe a&nbsp; '''ring'''?
 |type="[]"}
 - Table &nbsp;$\rm A3$,
@@ Line 54: / Line 54: @@
 - Table &nbsp;$\rm A3$&nbsp; and table &nbsp;$\rm M4$&nbsp; together.
-{Which of the tables describe a field or a Galois field?
+{Which of the tables describe a&nbsp; '''(Galois) field'''?
 |type="[]"}
 - Table &nbsp;$\rm A3$,
@@ Line 61: / Line 61: @@
 - Table &nbsp;$\rm A4$&nbsp; and table &nbsp;$\rm M4$&nbsp; together.
-{Which elements of the set&nbsp; $\{0, \, 1, \, 2\} \ \Rightarrow \ q = 3$&nbsp; are primitive?
+{Which elements of the set&nbsp; $\{0, \, 1, \, 2\} \ \Rightarrow \ q = 3$&nbsp; are&nbsp; "primitive"?
 |type="[]"}
 - $z_0 = 0$,
@@ Line 67: / Line 67: @@
 + $z_2 = 2.$
-{Which elements of the set&nbsp; $\{0, \, 1, \, 2, \, 3\} \ \Rightarrow \ q = 4$&nbsp; are primitive?
+{Which elements of the set&nbsp; $\{0, \, 1, \, 2, \, 3\} \ \Rightarrow \ q = 4$&nbsp; are&nbsp; "primitive"?
 |type="[]"}
 - $z_0 = 0$,

← Older revision		Revision as of 14:41, 24 August 2022
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−	[[Category:Channel Coding: Exercises\|^2.1 ~~Einige Grundlagen der~~ Algebra^]]	+	[[Category:Channel Coding: Exercises\|^2.1 Some Basics of Algebra^]]

@@ Line 30: / Line 30: @@
 Hints:
-* The exercise covers the topic of the chapter&nbsp; [[Channel_Coding/Some_Basics_of_Algebra| some basics of algebra]].
+* The exercise covers the topic of the chapter&nbsp; [[Channel_Coding/Some_Basics_of_Algebra| "some basics of algebra"]].
 * Note that for group, ring and field with each&nbsp; $q$&nbsp; elements, the arithmetic operations "$+$" and "$\hspace{0.05cm}\cdot\hspace{0.05cm}$" are to be understood modulo&nbsp; $q$&nbsp; respectively.
@@ Line 108: / Line 108: @@
-For example, a Galois field $\rm GF(4)$ is obtained by extending the binary set $\{0, \, 1\}$ to the set $\{0, \, 1, \, \alpha, \, 1+\alpha\}$. For more details, see the [[Channel_Coding/Extension_Field#GF.2822.29_.E2.80.93_Example_of_an_extension.C3.B6rpers|Example of an extension field]] page.
+For example, a Galois field $\rm GF(4)$ is obtained by extending the binary set $\{0, \, 1\}$ to the set $\{0, \, 1, \, \alpha, \, 1+\alpha\}$. For more details, see the [[Channel_Coding/Extension_Field#GF.2822.29_.E2.80.93_Example_of_an_extension.C3.B6rpers|"example of an extension field"]] page.

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Kanalcodierung/Einige Grundlagen der Algebra}}
-[[File:P_ID2490__KC_A_2_1.png|right|frame|"Addition" und "Multiplikation" für&nbsp; $q = 3$&nbsp; und&nbsp; $q = 4$]]
+[[File:EN_KC_A_2_1.png|right|frame|"Addition" und "Multiplikation" für&nbsp; $q = 3$&nbsp; und&nbsp; $q = 4$]]
 Im&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra|Theorieteil]]&nbsp; zu diesem Kapitel wurden verschiedene algebraische Begriffe definiert. Für das Folgende setzen wir voraus, dass alle Mengen aus jeweils&nbsp; $q$&nbsp; Elementen bestehen, wobei hier entweder&nbsp; $q = 3$&nbsp; oder&nbsp; $q = 4$&nbsp; gelten soll. Dann gilt:
 * Eine&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra#Definition_und_Beispiele_einer_algebraischen_Gruppe|algebraische Gruppe]]&nbsp; ist eine endliche Menge&nbsp; $G = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; zusammen mit einer zwischen allen Elementen definierten Verknüpfungsvorschrift. Eine additive Gruppe wird mit&nbsp; $(G, \ +)$&nbsp; bezeichnet, eine multiplikative mit&nbsp; $(G, \ \cdot)$.

← Older revision	Revision as of 14:36, 24 August 2022
(No difference)

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Kanalcodierung/Einige Grundlagen der Algebra}}
-[[File:P_ID2490__KC_A_2_1.png|right|frame|&bdquo;Addition" und &bdquo;Multiplikation" für&nbsp; $q = 3$&nbsp; und&nbsp; $q = 4$]]
+[[File:P_ID2490__KC_A_2_1.png|right|frame|"Addition" und "Multiplikation" für&nbsp; $q = 3$&nbsp; und&nbsp; $q = 4$]]
 Im&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra|Theorieteil]]&nbsp; zu diesem Kapitel wurden verschiedene algebraische Begriffe definiert. Für das Folgende setzen wir voraus, dass alle Mengen aus jeweils&nbsp; $q$&nbsp; Elementen bestehen, wobei hier entweder&nbsp; $q = 3$&nbsp; oder&nbsp; $q = 4$&nbsp; gelten soll. Dann gilt:
 * Eine&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra#Definition_und_Beispiele_einer_algebraischen_Gruppe|algebraische Gruppe]]&nbsp; ist eine endliche Menge&nbsp; $G = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; zusammen mit einer zwischen allen Elementen definierten Verknüpfungsvorschrift. Eine additive Gruppe wird mit&nbsp; $(G, \ +)$&nbsp; bezeichnet, eine multiplikative mit&nbsp; $(G, \ \cdot)$.
-* Ein&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra#Definition_und_Beispiele_eines_algebraischen_Rings|algebraischer Ring]]&nbsp; kennzeichnet eine Menge&nbsp; $R = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; zusammen mit zwei darin definierten Rechenoperationen, nämlich der Addition&nbsp; (&bdquo;$+$")&nbsp; und der Multiplikation&nbsp; (&bdquo;$\hspace{0.05cm}\cdot\hspace{0.05cm}$").
+* Ein&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra#Definition_und_Beispiele_eines_algebraischen_Rings|algebraischer Ring]]&nbsp; kennzeichnet eine Menge&nbsp; $R = \{0, \, 1, \hspace{0.05cm}\text{...} \hspace{0.1cm} , \ q-1\}$&nbsp; zusammen mit zwei darin definierten Rechenoperationen, nämlich der Addition&nbsp; ("$+$")&nbsp; und der Multiplikation&nbsp; ("$\hspace{0.05cm}\cdot\hspace{0.05cm}$").
 * Ein&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra#Gruppe.2C_Ring.2C_K.C3.B6rper_.E2.80.93_algebraische_Grundbegriffe|algebraischer Körper]]&nbsp; ist ein Ring, bei dem zusätzlich die Division erlaubt ist und das Kommutativgesetz erfüllt wird.
@@ Line 31: / Line 31: @@
 ''Hinweise:''
 * Die Aufgabe behandelt das Themengebiet des Kapitels&nbsp; [[Channel_Coding/Einige_Grundlagen_der_Algebra| Einige Grundlagen der Algebra]].
-* Beachten Sie, dass bei Gruppe, Ring und Körper mit jeweils&nbsp; $q$&nbsp; Elementen die Rechenoperationen &bdquo;$+$" und &bdquo;$\hspace{0.05cm}\cdot\hspace{0.05cm}$" jeweils modulo&nbsp; $q$&nbsp; zu verstehen sind.
+* Beachten Sie, dass bei Gruppe, Ring und Körper mit jeweils&nbsp; $q$&nbsp; Elementen die Rechenoperationen "$+$" und "$\hspace{0.05cm}\cdot\hspace{0.05cm}$" jeweils modulo&nbsp; $q$&nbsp; zu verstehen sind.
@@ Line 105: / Line 105: @@
 *Dagegen führen die Tabellen A4 (Addition) und M4 Multiplikation) zusammen mit der Menge $\{0, \, 1, \, 2, \, 3\}$ nicht zum Galoisfeld ${\rm GF}(4)$.
 *Eine Bedingung für ein Galoisfeld ist, dass es für jedes Element $z_i$ eine multiplikative Inverse ${\rm Inv}_{\rm M}{(z_i)}$ gibt, so dass die Gleichung $z_i \cdot {\rm Inv}_{\rm M}(z_i) = 1$ erfüllt ist.
-*Nach Tabelle M4 existiert $\rm Inv_M(2)$ aber nicht. In der dritten Zeile gibt es keine &bdquo;$1$".
+*Nach Tabelle M4 existiert $\rm Inv_M(2)$ aber nicht. In der dritten Zeile gibt es keine "$1$".

← Older revision		Revision as of 12:49, 23 March 2021
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−	[[Category:~~Aufgaben zu Kanalcodierung~~\|^2.1 Einige Grundlagen der Algebra^]]	+	[[Category:Channel Coding: Exercises\|^2.1 Einige Grundlagen der Algebra^]]