Difference between revisions of "Aufgaben:Exercise 2.2: Properties of Galois Fields"

Latest revision as of 15:11, 28 August 2022

Addition / multiplication for $q = 5$ and $q = 6$

Here we consider the sets of numbers

$Z_5 = \{0, \, 1, \, 2, \, 3, \, 4\} \ \Rightarrow \ q = 5$,

$Z_6 = \{0, \, 1, \, 2, \, 3, \, 4,\, 5\} \ \Rightarrow \ q = 6$.

In the adjacent graph, the (partially incomplete) addition and multiplication tables for $q = 5$ and $q = 6$ are given, where both addition ("$+$") and multiplication ("$\hspace{0.05cm}\cdot\hspace{0.05cm}$") modulo $q$ are to be understood.

To be checked is whether the number sets $Z_5$ and $Z_6$ satisfy all the conditions of a Galois field $\rm GF(5)$ and $\rm GF(6)$, respectively.

In the "theory section" a total of eight conditions are mentioned, all of which must be met. You are to check only two of these conditions:

$\rm(D)$ For all elements there is an additive inverse (Inverse for "$+$"):

$$\forall \hspace{0.15cm} z_i \in {\rm GF}(q),\hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_A}(z_i) \in {\rm GF}(q)\text{:}\hspace{0.5cm}z_i + {\rm Inv_A}(z_i) = 0 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_A}(z_i) = -z_i \hspace{0.05cm}.$$

$\rm(E)$ All elements have a multiplicative inverse (Inverse for "$\hspace{0.05cm}\cdot\hspace{0.05cm}$"):

$$\forall \hspace{0.15cm} z_i \in {\rm GF}(q),\hspace{0.15cm} z_i \ne 0, \hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_M}(z_i) \in {\rm GF}(q)\text{:}\hspace{0.5cm}z_i \cdot {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = z_i^{-1}\hspace{0.05cm}.$$

The other conditions for a Galois field, viz.

Closure,
Existence of zero– and identity element,
validity of commutative law, associative law and distributive law

are satisfied by both, $Z_5$ and $Z_6$.

Hints: The exercise refers to the chapter "Some Basics of Algebra".

Questions

$A_{04} \ = \ $

$A_{14} \ = \ $

$A_{44} \ = \ $

$M_{04} \ = \ $

$M_{14} \ = \ $

$M_{44} \ = \ $

	Yes.
	No, there is not an additive inverse for all elements $(0, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 4)$ .
	No, the elements $1, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 4$ do not all have a multiplicative inverse.

	Yes.
	No, there is not an additive inverse for all elements $(0, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 5)$ .
	No, the elements $1, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 5$ do not all have a multiplicative inverse.

	$Z_{10} = \{0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9\}$ is a Galois field?
	$Z_{11} = \{0, \, 1, \, 2, \, 3, \, 4, \,5, \, 6, \, 7, \, 8, \, 9, \, 10\}$ is a Galois field?
	$Z_{12} = \{0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9, \, 10, \, 11\}$ is a Galois field?

Solution

(1) In general, for $0 ≤ \mu ≤ 4 \text{:} \hspace{0.2cm} A_{\mu 4} = (\mu + 4) \, {\rm mod} \, 5$. It follows:

$$A_{04} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (0+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 4}\hspace{0.05cm},\hspace{0.2cm}A_{14}=(1+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}A_{24}=(2+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.05cm},$$

$$A_{34} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (3+4)\hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5= 2\hspace{0.05cm},\hspace{0.2cm}A_{44}=(4+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 3}\hspace{0.05cm}.$$

Due to the commutative law of addition,

$$z_i + z_j = z_j + z_i \hspace{0.5cm} {\rm for \hspace{0.2cm}all\hspace{0.2cm} } z_i, z_j \in Z_5\hspace{0.05cm},$$

the last column of the addition table is of course identical to the last row of the same table.

(2) Now $M_{\mu 4} = (\mu \cdot 4) \, {\rm mod} \, 5$ and we obtain:

$$M_{04} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (0\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}M_{14}=(1\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 4}\hspace{0.05cm},\hspace{0.2cm}M_{24}=(2\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3\hspace{0.05cm},$$

$$M_{34} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (3\cdot4)\hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2\hspace{0.05cm},\hspace{0.2cm}M_{44}=(4\cdot 4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 1}\hspace{0.05cm}.$$

Since multiplication is also commutative, the last column in the multiplication table again matches the last row.

Addition/multiplication tables for $q = 5$

(3) The graph shows the full addition and multiplication tables for $q = 5$. You can see:

In the addition table there is exactly one zero in each row (and also in each column).

So for every $z_i ∈ Z_5$ there is an ${\rm Inv}_{\rm A} (z_i)$ that satisfies the condition $[z_i + {\rm Inv}_{\rm A}(z_i)] \, {\rm mod} \, 5 = 0$:

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 0\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = 0 \hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-1) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 4 \hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 2\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-2) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3 \hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 3\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-3) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2 \hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 4\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm}.$$

In the multiplication table we leave the zero element (first row and first column) out of consideration.

In all other rows and columns of the lower table there is indeed exactly one each.

From the condition $[z_i \cdot {\rm Inv}_{\rm M}(z_i)] \, {\rm mod} \, 5 = 1$ one obtains:

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} z_i \cdot {\rm Inv_M}(z_i) = 1\hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 2 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 3 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} z_i \cdot {\rm Inv_M}(z_i) = 6 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 3 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 2 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} z_i \cdot {\rm Inv_M}(z_i) = 6 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 4 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 4 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} z_i \cdot {\rm Inv_M}(z_i) = 16 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm}.$$

Since both the required additive and multiplicative inverses exist ⇒ $Z_5$ describes a Galois field $\rm GF(5)$

Correct is the proposed solution 1.

(4) From the blue addition table on the statement page, we see that all numbers $(0, \, 1, \, 2, \, 3, \, 4, \, 5)$ of the set $Z_6$ have an additive inverse

⇒ in each row (and column) there is exactly one zero.

On the other hand, a multiplicative inverse ${\rm Inv}_{\rm M}(z_i)$ exists only for $z_i = 1$ and $z_i = 5$, viz.

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} z_i \cdot {\rm Inv_M}(z_i) = 1\hspace{0.05cm},$$

$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 5 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 5 \hspace{0.25cm} \Rightarrow \hspace{0.25cm} z_i \cdot {\rm Inv_M}(z_i) = 25 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 6 = 1 \hspace{0.05cm}.$$

For $z_i = 2, \ z_i = 3$ and $z_i = 4$, we find no element $z_j$, so that $(z_i \cdot z_j) \, {\rm mod} \, 6 = 1$.

Correct is the proposed solution 3 ⇒ the blue tables for $q = 6$ do not yield a Galois field $\rm GF(6)$.

(5) Correct is the proposed solution 2:

A finite number set $Z_q = \{0, \, 1, \hspace{0.05cm} \text{...} \hspace{0.1cm} , \, q-1\}$ of natural numbers leads to a Galois field only if $q$ is a prime number.

Of the number sets mentioned above, this is true only for $Z_{11}$.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Kanalcodierung/Einige Grundlagen der Algebra}}
+{{quiz-Header|Buchseite=Channel_Coding/Some_Basics_of_Algebra}}
-[[File:P_ID2492__KC_A_2_2.png|right|frame|Tabellen zur Addition und Multiplikation für $q = 5$ und $q = 6$]]
+[[File:EN_KC_A_2_2.png|right|frame|Addition / multiplication for&nbsp; $q = 5$&nbsp; and&nbsp; $q = 6$]]
-Wir betrachten hier die Zahlenmengen
+Here we consider the sets of numbers
 * $Z_5 = \{0, \, 1, \, 2, \, 3, \, 4\} \ \Rightarrow \ q = 5$,
 * $Z_6 = \{0, \, 1, \, 2, \, 3, \, 4,\, 5\} \ \Rightarrow \ q = 6$.
-In nebenstehender Grafik sind die (teilweise unvollständigen) Additions&ndash; und Multiplikationstabellen für $q = 5$ und $q = 6$ angegeben, wobei sowohl die Addition (&bdquo;$+$&rdquo;) als auch die Multiplikation (&bdquo;$\cdot$&rdquo;) modulo $q$ zu verstehen sind.
+In the adjacent graph,&nbsp;  the (partially incomplete)&nbsp; addition and multiplication tables for&nbsp; $q = 5$&nbsp; and&nbsp; $q = 6$&nbsp; are given,&nbsp; where both addition&nbsp; ("$+$")&nbsp; and multiplication&nbsp; ("$\hspace{0.05cm}\cdot\hspace{0.05cm}$")&nbsp; modulo&nbsp; $q$&nbsp; are to be understood.
+To be checked is whether the number sets&nbsp; $Z_5$&nbsp; and&nbsp; $Z_6$&nbsp; satisfy all the conditions of a Galois field&nbsp; $\rm GF(5)$&nbsp; and&nbsp; $\rm GF(6)$,&nbsp; respectively.
-Zu überprüfen ist, ob die Zahlenmengen $Z_5$ und $Z_6$ alle Bedingungen eines Galoisfeldes $\rm GF(5)$ bzw. $\rm GF(6)$ erfüllen. Im [[Kanalcodierung/Einige_Grundlagen_der_Algebra#Definition_eines_Galoisfeldes|Theorieteil]] werden insgesamt acht Bedingungen genannt, die alle erfüllt sein müssen. Sie sollen nur zwei dieser Bedingungen überprüfen:
+In the&nbsp; [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field|"theory section"]]&nbsp; a total of eight conditions are mentioned,&nbsp; all of which must be met.&nbsp; You are to check only two of these conditions:
-'''(D)'''&nbsp;  Für alle Elemente gibt es eine <b>additive Inverse</b> (<i>Inverse for &bdquo;$+$&rdquo;</i>):
+$\rm(D)$&nbsp;  For all elements there is an&nbsp; <b>additive inverse</b>&nbsp; (Inverse&nbsp; for&nbsp; "$+$"):
-:$$\forall \hspace{0.15cm}  z_i \in {\rm GF}(q),\hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_A}(z_i) \in {\rm GF}(q)\text{:}$$
+:$$\forall \hspace{0.15cm}  z_i \in {\rm GF}(q),\hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_A}(z_i) \in {\rm GF}(q)\text{:}\hspace{0.5cm}z_i + {\rm Inv_A}(z_i) = 0  \hspace{0.25cm} \Rightarrow \hspace{0.25cm}
-::$$\hspace{0.25cm}z_i + {\rm Inv_A}(z_i) = 0  \hspace{0.25cm} \Rightarrow \hspace{0.25cm}
 {\rm Inv_A}(z_i) = -z_i \hspace{0.05cm}.$$
-'''(E)'''&nbsp;  Alle Elemente haben eine <b>multiplikative Inverse</b> (<i>Inverse for &bdquo;$\cdot$&rdquo;</i>):
+$\rm(E)$&nbsp;  All elements have a&nbsp; <b>multiplicative inverse</b>&nbsp; (Inverse&nbsp; for&nbsp; "$\hspace{0.05cm}\cdot\hspace{0.05cm}$"):
-:$$\forall \hspace{0.15cm}  z_i \in {\rm GF}(q),\hspace{0.15cm} z_i \ne 0, \hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_M}(z_i) \in {\rm GF}(q)\text{:}$$
+:$$\forall \hspace{0.15cm}  z_i \in {\rm GF}(q),\hspace{0.15cm} z_i \ne 0, \hspace{0.15cm} \exists \hspace{0.15cm} {\rm Inv_M}(z_i) \in {\rm GF}(q)\text{:}\hspace{0.5cm}z_i \cdot {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}
-::$$\hspace{0.25cm}z_i \cdot {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}
 {\rm Inv_M}(z_i) = z_i^{-1}\hspace{0.05cm}.$$
-Die weiteren Bedingungen für ein Galoisfeld, nämlich
+The other conditions for a Galois field, viz.
 * Closure,
-* Existenz von Null&ndash; und Einselelement,
+* Existence of zero&ndash; and identity element,
-* Gültigkeit von Kommutativ&ndash;, Assoziativ&ndash; und Distributivgesetz
+* validity of commutative law, associative law and distributive law
-werden sowohl von $Z_5$ als auch von $Z_6$ erfüllt.
+are satisfied by both,&nbsp; $Z_5$&nbsp; and&nbsp; $Z_6$.
@@ Line 33: / Line 34: @@
-''Hinweise:''
+Hints:&nbsp; The exercise refers to the chapter&nbsp; [[Channel_Coding/Some_Basics_of_Algebra| "Some Basics of Algebra"]].
-* Die Aufgabe bezieht sich auf das Kapitel [[Kanalcodierung/Einige_Grundlagen_der_Algebra| Einige Grundlagen der Algebra]].
-* Sollte die Eingabe des Zahlenwertes &bdquo;0&rdquo; erforderlich sein, so geben Sie bitte &bdquo;0.&rdquo; ein.
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Ergänzen Sie die Additionstabelle für $q = 5$. Geben Sie folgende Werte ein:
+{Complete the addition table for&nbsp; $q = 5$.&nbsp; Enter the following values:
 |type="{}"}
-$A_{04} \ = \ ${ 4 3% }
+$A_{04} \ = \ ${ 4 }
-$A_{14} \ = \ ${ 0 3% }
+$A_{14} \ = \ ${ 0. }
-$A_{44} \ = \ ${ 3 3% }
+$A_{44} \ = \ ${ 3 }
-{Ergänzen Sie die Multiplikationstabelle für $q = 5$. Geben Sie folgende Werte ein:
+{Complete the multiplication table for&nbsp; $q = 5$.&nbsp; Enter the following values:
 |type="{}"}
-$M_{04} \ = \ ${ 0 3% }
+$M_{04} \ = \ ${ 0. }
-$M_{14} \ = \ ${ 4 3% }
+$M_{14} \ = \ ${ 4. }
-$M_{44} \ = \ ${ 1 3% }
+$M_{44} \ = \ ${ 1. }
-{Erfüllt die Menge $Z_5$ die Bedingungen eines Galoisfeldes?
+{Does the&nbsp; $Z_5$&nbsp; set satisfy the conditions of a Galois field?
 |type="[]"}
-+ Ja.
++ Yes.
-- Nein, es gibt nicht für alle Elemente $(0, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 4)$ eine additive Inverse.
+- No,&nbsp; there is not an additive inverse for all elements&nbsp; $(0, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 4)$&nbsp;.
-- Nein, die Elemente $1, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 4$ haben nicht alle eine multiplikative Inverse.
+- No,&nbsp; the elements&nbsp; $1, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 4$&nbsp; do not all have a multiplicative inverse.
-{Erfüllt die Menge $Z_6$ die Bedingungen eines Galoisfeldes?
+{Does the&nbsp; $Z_6$&nbsp; set satisfy the conditions of a Galois field?
 |type="[]"}
-- Ja.
+- Yes.
-- Nein, es gibt nicht für alle Elemente $(0, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 5)$ eine additive Inverse.
+- No,&nbsp; there is not an additive inverse for all elements&nbsp; $(0, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 5)$&nbsp;.
-+ Nein, die Elemente $1, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 5$ haben nicht alle eine multiplikative Inverse.
++ No,&nbsp; the elements&nbsp; $1, \hspace{0.05cm}\text{...} \hspace{0.1cm}, 5$&nbsp; do not all have a multiplicative inverse.
-{Die Zahlenmengen $Z_2, \ Z_3, \ Z_5$ und $Z_7$ ergeben ein Galoisfeld, die Mengen $Z_4, \ Z_6, \ Z_8, \ Z_9$ dagegen nicht. Was folgern Sie daraus?
+{The number sets&nbsp; $Z_2, \ Z_3, \ Z_5$&nbsp; and $Z_7$&nbsp; yield a Galois field,&nbsp; but the sets&nbsp; $Z_4, \ Z_6, \ Z_8, \ Z_9$&nbsp; do not.&nbsp; What do you conclude from this?
 |type="[]"}
-- $Z_{10} = \{0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9\}$ ist ein Galoisfeld?
+- $Z_{10} = \{0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9\}$ is a Galois field?
-+ $Z_{11} = \{0, \, 1, \, 2, \, 3, \, 4, \,5, \, 6, \, 7, \, 8, \, 9, \, 10\}$ ist ein Galoisfeld?
++ $Z_{11} = \{0, \, 1, \, 2, \, 3, \, 4, \,5, \, 6, \, 7, \, 8, \, 9, \, 10\}$ is a Galois field?
-- $Z_{12} = \{0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9, \, 10, \, 11\}$ ist ein Galoisfeld?
+- $Z_{12} = \{0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9, \, 10, \, 11\}$ is a Galois field?
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Allgemein gilt für $0 &#8804; \mu &#8804; 4 \text{:} \hspace{0.2cm} A_{\mu 4} = (\mu + 4) \, {\rm mod} \, 5$. Daraus folgt:
+'''(1)'''&nbsp;  In general,&nbsp; for&nbsp; $0 &#8804; \mu &#8804; 4 \text{:} \hspace{0.2cm} A_{\mu 4} = (\mu + 4) \, {\rm mod} \, 5$.&nbsp; It follows:
-:$$A_{04} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (0+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 4}\hspace{0.05cm},\hspace{0.2cm}A_{14}=(1+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}A_{24}=(2+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 1}\hspace{0.05cm},$$
+:$$A_{04} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (0+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 4}\hspace{0.05cm},\hspace{0.2cm}A_{14}=(1+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}A_{24}=(2+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.05cm},$$
-:$$A_{34} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (3+4)\hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 2}\hspace{0.05cm},\hspace{0.2cm}A_{44}=(4+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 3}\hspace{0.05cm}.$$
+:$$A_{34} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (3+4)\hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5= 2\hspace{0.05cm},\hspace{0.2cm}A_{44}=(4+4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 3}\hspace{0.05cm}.$$
+Due to the commutative law of addition,
+:$$z_i + z_j = z_j + z_i \hspace{0.5cm} {\rm for  \hspace{0.2cm}all\hspace{0.2cm} } z_i, z_j \in Z_5\hspace{0.05cm},$$
-Aufgrund des Kommutativgesetzes der Addition,
+the last column of the addition table is of course identical to the last row of the same table.
-:$$z_i + z_j = z_j + z_i \hspace{0.5cm} {\rm f\ddot{u}r  \hspace{0.2cm}alle\hspace{0.2cm} } z_i, z_j \in Z_5\hspace{0.05cm},$$
-ist natürlich die letzte Spalte der Additionstabelle identisch mit der letzten Zeile der gleichen Tabelle.
-'''(2)'''&nbsp; Nun gilt $M_{\mu 4} = (\mu \cdot 4) \, {\rm mod} \, 5$ und man erhält:
+'''(2)'''&nbsp; Now $M_{\mu 4} = (\mu \cdot 4) \, {\rm mod} \, 5$&nbsp; and we obtain:
-:$$M_{04} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (0\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}M_{14}=(1\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 4}\hspace{0.05cm},\hspace{0.2cm}M_{24}=(2\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 3}\hspace{0.05cm},$$
+:$$M_{04} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (0\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}M_{14}=(1\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 4}\hspace{0.05cm},\hspace{0.2cm}M_{24}=(2\cdot4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3\hspace{0.05cm},$$
-:$$M_{34} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (3\cdot4)\hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 2}\hspace{0.05cm},\hspace{0.2cm}M_{44}=(4\cdot 4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 1}\hspace{0.05cm}.$$
+:$$M_{34} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} (3\cdot4)\hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2\hspace{0.05cm},\hspace{0.2cm}M_{44}=(4\cdot 4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 \hspace{0.15cm}\underline{= 1}\hspace{0.05cm}.$$
-Da die Multiplikation ebenfalls kommutativ ist, stimmt auch in der Multiplikationstabelle die letzte Spalte wieder mit der letzten Zeile überein.
+Since multiplication is also commutative,&nbsp; the last column in the multiplication table again matches the last row.
-'''(3)'''&nbsp; [[File:P_ID2493__KC_A_2_2c.png|right|frame|Additions– und Multiplikationstabelle für $q = 5$]] Die Grafik zeigt die vollständigen Additions&ndash; und Multiplikationstabellen für $q = 5$. Man erkennt:
-* In der Additionstabelle gibt es in jeder Zeile (und auch in jeder Spalte) genau eine Null. Zu jedem $z_i &#8712; Z_5$ gibt es also ein ${\rm Inv}_{\rm A} (z_i)$, das die Bedingung $[z_i + {\rm Inv}_{\rm A}(z_i)] \, {\rm mod} \, 5 = 0$ erfüllt:
-:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 0: \hspace{0.1cm}{\rm Inv_A}(z_i) = 0  \hspace{0.05cm},$$
-:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1:\hspace{0.1cm}{\rm Inv_A}(z_i) = (-1) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 4 \hspace{0.05cm},$$
-:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 2: \hspace{0.1cm}{\rm Inv_A}(z_i) = (-2) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3 \hspace{0.05cm},$$
-:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 3: \hspace{0.1cm}{\rm Inv_A}(z_i) = (-3) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2 \hspace{0.05cm},$$
-:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 4: \hspace{0.1cm}{\rm Inv_A}(z_i) = (-4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm}.$$
-* In der Multiplikationstabelle lassen wir das Nullelement (erste Zeile und erste Spalte) außer Betracht. In allen anderen Zeilen und Spalten der unteren Tabelle gibt es tatsächlich jeweils genau eine Eins. Aus der Bedingung $[z_i \cdot {\rm Inv}_{\rm M}(z_i)] \, {\rm mod} \, 5 = 1$ erhält man:
+[[File:P_ID2493__KC_A_2_2c.png|right|frame|Addition/multiplication tables for&nbsp; $q = 5$]]
+'''(3)'''&nbsp; The graph shows the full addition and multiplication tables for&nbsp; $q = 5$.&nbsp; You can see:
+* In the addition table there is exactly one zero in each row&nbsp; (and also in each column).&nbsp;
+*So for every&nbsp; $z_i &#8712; Z_5$&nbsp; there is an&nbsp; ${\rm Inv}_{\rm A} (z_i)$&nbsp; that satisfies the condition&nbsp; $[z_i + {\rm Inv}_{\rm A}(z_i)] \, {\rm mod} \, 5 = 0$:
+:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 0\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = 0  \hspace{0.05cm},$$
+:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-1) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 4 \hspace{0.05cm},$$
+:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 2\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-2) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3 \hspace{0.05cm},$$
+:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 3\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-3) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2 \hspace{0.05cm},$$
+:$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 4\hspace{0.25cm} \Rightarrow \hspace{0.25cm}{\rm Inv_A}(z_i) = (-4) \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm}.$$
+* In the multiplication table we leave the zero element&nbsp; (first row and first column)&nbsp; out of consideration.
+*In all other rows and columns of the lower table there is indeed exactly one each.&nbsp;
+*From the condition $[z_i \cdot {\rm Inv}_{\rm M}(z_i)] \, {\rm mod} \, 5 = 1$&nbsp; one obtains:
 :$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1  \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}  z_i \cdot {\rm Inv_M}(z_i) =  1\hspace{0.05cm},$$
 :$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 2  \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 3 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}  z_i \cdot {\rm Inv_M}(z_i) =  6 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm},$$
@@ Line 106: / Line 115: @@
 :$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 4  \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 4 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}  z_i \cdot {\rm Inv_M}(z_i) =  16 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1 \hspace{0.05cm}.$$
-Da sowohl die erforderlichen additiven als auch die multiplikativen Inversen existieren beschreibt $Z_5$ ein Galoisfeld $\rm GF(5)$ &nbsp;&#8658;&nbsp; Richtig ist der <u>Lösungsvorschlag 1</u>.
+*Since both the required additive and multiplicative inverses exist &nbsp; &rArr; &nbsp; $Z_5$ describes a Galois field $\rm GF(5)$ &nbsp;
+*Correct is the <u>proposed solution 1</u>.
+'''(4)'''&nbsp; From the blue addition table on the statement page,&nbsp; we see that all numbers&nbsp; $(0, \, 1, \, 2, \, 3, \, 4, \, 5)$&nbsp; of the set $Z_6$&nbsp; have an additive inverse
-'''(4)'''&nbsp; Aus der blauen Additionstabelle auf der Angabenseite erkennt man, dass alle Zahlen $0, \, 1, \, 2, \, 3, \, 4, \, 5$ der Menge $Z_6$ eine additive Inverse besitzen &nbsp;&#8658;&nbsp; in jeder Zeile (und Spalte) gibt es genau eine Null.
+&nbsp; &#8658; &nbsp; in each row&nbsp; (and column)&nbsp; there is exactly one zero.
-Eine multiplikative Inverse ${\rm Inv}_{\rm M}(z_i)$ gibt es dagegen nur für $z_i = 1$ und $z_i = 5$, nämlich
+*On the other hand,&nbsp; a multiplicative inverse&nbsp; ${\rm Inv}_{\rm M}(z_i)$&nbsp; exists only for&nbsp; $z_i = 1$&nbsp; and&nbsp; $z_i = 5$,&nbsp; viz.
 :$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 1  \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 1 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}  z_i \cdot {\rm Inv_M}(z_i) =  1\hspace{0.05cm},$$
 :$$z_i \hspace{-0.1cm} \ = \ \hspace{-0.1cm} 5  \hspace{0.25cm} \Rightarrow \hspace{0.25cm} {\rm Inv_M}(z_i) = 5 \hspace{0.25cm} \Rightarrow \hspace{0.25cm}  z_i \cdot {\rm Inv_M}(z_i) =  25 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 6 = 1 \hspace{0.05cm}.$$
-Für $z_i = 2, \ z_i = 3$ und $z_i = 4$ findet man dagegen kein Element $z_j$, so dass $(z_i \cdot z_j) \, {\rm mod} \, 6 = 1$ ergibt. Richtig ist somit der <u>Lösungsvorschlag 3</u>. Die blauen Tabellen für $q = 6$ ergeben kein Galoisfeld $\rm GF(6)$.
+*For&nbsp; $z_i = 2, \ z_i = 3$ and $z_i = 4$,&nbsp; we find no element&nbsp; $z_j$,&nbsp; so that&nbsp; $(z_i \cdot z_j) \, {\rm mod} \, 6 = 1$.
+*Correct is the <u>proposed solution 3</u> &nbsp; &rArr; &nbsp; the blue tables for&nbsp; $q = 6$&nbsp; do not yield a Galois field&nbsp; $\rm GF(6)$.
-'''(5)'''&nbsp; Eine endliche Zahlenmenge $Z_q = \{0, \, 1, \, ... \, , \, q-1\}$ natürlicher Zahlen führt nur dann zu einem endlichen Zahlenkörper (dies ist die deutsche Bezeichnung für ein Galoisfeld), wenn $q$ eine Primzahl ist. Von den oben genannten Zahlenmengen trifft dies nur auf $Z_{11}$ zu &nbsp;&#8658;&nbsp; <u>Lösungsvorschlag 2</u>.
+'''(5)'''&nbsp; Correct is the&nbsp; <u>proposed solution 2</u>:
+*A finite number set&nbsp; $Z_q = \{0, \, 1, \hspace{0.05cm} \text{...} \hspace{0.1cm} , \, q-1\}$&nbsp; of natural numbers leads to a Galois field only if&nbsp; $q$&nbsp; is a prime number.
+*Of the number sets mentioned above,&nbsp; this is true only for&nbsp; $Z_{11}$.
 {{ML-Fuß}}
-[[Category:Aufgaben zu  Kanalcodierung|^2.1 Einige Grundlagen der Algebra^]]
+[[Category:Channel Coding: Exercises|^2.1 Some Basics of Algebra^]]