Aufgaben:Exercise 2.2Z: Galois Field GF(5) - Revision history

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	[[Category:Channel Coding: Exercises\|^2.1 Some Basics of Algebra^]]		[[Category:Channel Coding: Exercises\|^2.1 Some Basics of Algebra^]]
	[[de:Aufgaben:2.~~2Z_Galoisfeld_GF~~(5)]]		[[de:Aufgaben:Aufgabe 2.2Z: Galoisfeld GF(5)]]

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	*So you get as a final result:		*So you get as a final result:
	:$$\rm a \hspace{0.15cm}\underline{= 3}\hspace{0.05cm},\hspace{0.2cm}b \hspace{0.15cm}\underline{= 2}\hspace{0.05cm},\hspace{0.2cm}		:$$\rm a \hspace{0.15cm}\underline{= 3}\hspace{0.05cm},\hspace{0.2cm}b \hspace{0.15cm}\underline{= 2}\hspace{0.05cm},\hspace{0.2cm}c \hspace{0.15cm}\underline{= 1}\hspace{0.05cm},\hspace{0.2cm}d \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}e \hspace{0.15cm}\underline{= 4}\hspace{0.05cm}.$$
	c \hspace{0.15cm}\underline{= 1}\hspace{0.05cm},\hspace{0.2cm}d \hspace{0.15cm}\underline{= 0}\hspace{0.05cm},\hspace{0.2cm}
	e \hspace{0.15cm}\underline{= 4}\hspace{0.05cm}.$$

	The graph shows the addition and multiplication table for this set of numbers.		The graph shows the addition and multiplication table for this set of numbers.
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	'''(7)'''  Concerning the primitive elements one obtains		'''(7)'''  Concerning the primitive elements one obtains
	:$$\rm a \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 3 \hspace{0.05cm},\hspace{0.2cm} a^2 = 9 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 4		:$$\rm a \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 3 \hspace{0.05cm},\hspace{0.2cm} a^2 = 9 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 4\hspace{0.05cm},\hspace{0.2cm} a^3 = 27 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2\hspace{0.05cm},\hspace{0.2cm} a^4 = 81 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm primitive}\hspace{0.05cm},$$
	\hspace{0.05cm},\hspace{0.2cm} a^3 = 27 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 2\hspace{0.05cm},\hspace{0.2cm} a^4 = 81 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm primitive}\hspace{0.05cm},$$		:$$\rm b \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 2 \hspace{0.05cm},\hspace{0.2cm} b^2 = 4\hspace{0.05cm},\hspace{0.2cm} b^3 = 8 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3\hspace{0.05cm},\hspace{0.2cm} b^4 = 16 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm primitive}\hspace{0.05cm},$$
	:$$\rm b \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 2 \hspace{0.05cm},\hspace{0.2cm} b^2 = 4		:$$\rm e \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 4 \hspace{0.05cm},\hspace{0.2cm} e^2 = 16 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.05cm},\hspace{0.2cm} e^3 = \hspace{0.05cm} ...\hspace{0.05cm}= 4\hspace{0.05cm},\hspace{0.2cm} e^4 =\hspace{0.05cm} ...\hspace{0.05cm} = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm not\hspace{0.15cm} primitive}\hspace{0.05cm}.$$
	\hspace{0.05cm},\hspace{0.2cm} b^3 = 8 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 3\hspace{0.05cm},\hspace{0.2cm} b^4 = 16 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm primitive}\hspace{0.05cm},$$
	:$$\rm e \hspace{-0.15cm} \ = \ \hspace{-0.15cm} 4 \hspace{0.05cm},\hspace{0.2cm} e^2 = 16 \hspace{0.1cm}{\rm mod} \hspace{0.1cm} 5 = 1
	\hspace{0.05cm},\hspace{0.2cm} e^3 = \hspace{0.05cm} ...\hspace{0.05cm}= 4\hspace{0.05cm},\hspace{0.2cm} e^4 =\hspace{0.05cm} ...\hspace{0.05cm} = 1\hspace{0.13cm} \Rightarrow \hspace{0.13cm}{\rm not\hspace{0.15cm} primitive}\hspace{0.05cm}.$$

	*From the set  $Z_5 = \{0, \, 1, \, 2, \, 3, \, 4\}$   ⇒   "$2$"  and  "$3$"  are primitive elements   ⇒   <u>solution suggestions 1 and 2</u>.		*From the set  $Z_5 = \{0, \, 1, \, 2, \, 3, \, 4\}$   ⇒   "$2$"  and  "$3$"  are primitive elements   ⇒   <u>solution suggestions 1 and 2</u>.

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← Older revision		Revision as of 14:31, 16 March 2026
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	[[Category:Channel Coding: Exercises\|^2.1 Some Basics of Algebra^]]		[[Category:Channel Coding: Exercises\|^2.1 Some Basics of Algebra^]]
			[[de:Aufgaben:2.2Z_Galoisfeld_GF(5)]]

Guenter at 15:04, 28 August 2022

2022-08-28T15:04:39Z

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	* an addition table modulo 5,		* an addition table modulo 5,

	* a multiplication table modulo 5,		* a multiplication table modulo 5.


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	~~Hints~~:  The exercise is related to the topic of the chapter  [[Channel_Coding/Some_Basics_of_Algebra\| "Some Basics of Algebra"]].		Hint:  The exercise is related to the topic of the chapter  [[Channel_Coding/Some_Basics_of_Algebra\| "Some Basics of Algebra"]].

Guenter at 14:41, 28 August 2022

2022-08-28T14:41:20Z

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Guenter at 13:52, 28 August 2022

2022-08-28T13:52:31Z

← Older revision		Revision as of 15:52, 28 August 2022
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	{{quiz-Header\|Buchseite=Channel_Coding/Some_Basics_of_Algebra}}		{{quiz-Header\|Buchseite=Channel_Coding/Some_Basics_of_Algebra}}

	[[File:P_ID2494__KC_Z_2_2.png\|right\|frame\|Addition/multiplication for  $\{a, \, b, \, c, \, d, \, e\}$]]		[[File:P_ID2494__KC_Z_2_2.png\|right\|frame\|Addition/multiplication for  $\rm \{a, \, b, \, c, \, d, \, e\}$]]
	As in the  [[Aufgaben:Exercise_2.2:_Properties_of_Galois_Fields\|Exercise 2.2]]  we consider a finite field of order  $q = 5$  and thus the Galois field		As in the  [[Aufgaben:Exercise_2.2:_Properties_of_Galois_Fields\|"Exercise 2.2"]]  we consider a finite field of order  $q = 5$  and thus the Galois field
	:$${\rm GF}(5) = \{{a}, { b},{c},{d},{e}\}\hspace{0.05cm}.$$		:$$\rm GF(5) = \{{a}, { b},{c},{d},{e}\}\hspace{0.05cm}.$$

	No further statements are made about the elements. They can be integers or any mathematical expressions.		No further statements are made about the elements. They can be integers or any mathematical expressions.
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	The Galois field is determined exclusively by		The Galois field is determined exclusively by
	* an addition table modulo 5,		* an addition table modulo 5,

	* a multiplication table modulo 5,		* a multiplication table modulo 5,


	The main properties of a Galois field are compiled on the  [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field\|"first theory page"]] ~~. Here~~ reference is made to		The main properties of a Galois field are compiled on the  [[Channel_Coding/Some_Basics_of_Algebra#Definition_of_a_Galois_field\|"first theory page"]].  There,  reference is made to
	* the commutative and distributive laws,		* the commutative and distributive laws,
	* the neutral elements of addition and multiplication,
			* the neutral elements of addition and multiplication,

	* the inverse elements of addition and multiplication, and		* the inverse elements of addition and multiplication, and

	* the determination of primitive elements.		* the determination of primitive elements.


	In the present example  $\beta$  would be a primitive element if  $\beta^2, \ \beta^3$  and  $\beta^4$  (in general:  $\beta^{q-1})$  yield the remaining elements of the Galois field  $\rm GF(5)$  except for the zero element.		In the present example  $\beta$  would be a primitive element if  $\beta^2, \ \beta^3$  and  $\beta^4$  $($in general:  $\beta^{q-1})$  yield the remaining elements of the Galois field  $\rm GF(5)$  except for the zero element.






	~~Hints:~~
	* The exercise is related to the topic of the chapter [[Channel_Coding/Some_Basics_of_Algebra\| "Some Basics of Algebra"]].
	* Please don't be confused that in the text the set  $ \{{a}, { b},{c},{d},{e}\}$  is used and in the tables the set  $ \rm \{{a}, { b},{c},{d},{e}\}$. Apologies.

			Hints:  The exercise is related to the topic of the chapter  [[Channel_Coding/Some_Basics_of_Algebra\| "Some Basics of Algebra"]].


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	{Determine the neutral element of addition.		{Determine the neutral element of addition.
	\|type="()"}		\|type="()"}
	- $N_{\rm A} = a$,		- $N_{\rm A} = \rm a$,
	- $N_{\rm A} = b$,		- $N_{\rm A} = \rm b$,
	- $N_{\rm A} = c$,		- $N_{\rm A} = \rm c$,
	+ $N_{\rm A} = d$,		+ $N_{\rm A} = \rm d$,
	- $N_{\rm A} = e$.		- $N_{\rm A} = \rm e$.

	{Determine the neutral element of multiplication.		{Determine the neutral element of multiplication.
	\|type="()"}		\|type="()"}
	- $N_{\rm M} = a$,		- $N_{\rm M} = \rm a$,
	- $N_{\rm M} = b$,		- $N_{\rm M} = \rm b$,
	+ $N_{\rm M} = c$,		+ $N_{\rm M} = \rm c$,
	- $N_{\rm M} = d$,		- $N_{\rm M} = \rm d$,
	- $N_{\rm M} = e$.		- $N_{\rm M} = \rm e$.

	{If the commutative law is satisfied,		{If the commutative law is satisfied,
	\|type="[]"}		\|type="[]"}
	+ with respect to addition, ~~for example~~  $a + b = b + a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d + e = e + d$,		+ with respect to addition,  e.g.  $\rm a + b = b + a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d + e = e + d$,
	+ with respect to multiplication, ~~for example~~  $a \cdot b = b \cdot a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d \cdot e = e \cdot d$.		+ with respect to multiplication,  e.g.  $\rm a \cdot b = b \cdot a, \hspace{0.05cm}\text{ ...} \hspace{0.1cm}, \ d \cdot e = e \cdot d$.

	{For which expressions is the distributive law satisfied?		{For which expressions is the distributive law satisfied?
	\|type="[]"}		\|type="[]"}
	+ $a \cdot (b + c) = a \cdot b + a \cdot c$,		+ $\rm a \cdot (b + c) = a \cdot b + a \cdot c$,
	+ $d \cdot (b + c) = d \cdot b + d \cdot c$,		+ $\rm d \cdot (b + c) = d \cdot b + d \cdot c$,
	+ $e \cdot (a + b) = e \cdot a + e \cdot b$.		+ $\rm e \cdot (a + b) = e \cdot a + e \cdot b$.

	{Replace  $a, \ b, \ c, \ d, \ e$  with elements of the number set  $\{0, \, 1, \, 2, \, 3, \, 4\}$ so that equal operation tables result.		{Replace  $\rm a, \ b, \ c, \ d, \ e$  with elements of the number set  $\{0, \, 1, \, 2, \, 3, \, 4\}$  so that equal operation tables result.
	\|type="{}"}		\|type="{}"}
	$a \hspace{0.2cm} = \ ${ 3 }		$\rm a \hspace{0.2cm} = \ ${ 3 }
	$b \hspace{0.25cm} = \ ${ 2 }		$\rm b \hspace{0.25cm} = \ ${ 2 }
	$c \hspace{0.25cm} = \ ${ 1 }		$\rm c \hspace{0.25cm} = \ ${ 1 }
	$d \hspace{0.2cm} = \ ${ 0. }		$\rm d \hspace{0.2cm} = \ ${ 0. }
	$e \hspace{0.2cm} = \ ${ 4 }		$\rm e \hspace{0.2cm} = \ ${ 4 }

	{What statements are true regarding inverse elements?		{What statements are true regarding inverse elements?
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	{Which of the elements are primitive?		{Which of the elements are primitive?
	\|type="[]"}		\|type="[]"}
	+ $a = 3$.		+ $\rm a = 3$.
	+ $b = 2$,		+ $\rm b = 2$,
	- $e = 4$.		- $\rm e = 4$.
	</quiz>		</quiz>

Noah at 20:31, 26 August 2022

2022-08-26T20:31:35Z

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	Hints:		Hints:
	* The exercise is related to the topic of the chapter [[Channel_Coding/Some_Basics_of_Algebra\| "Some Basics of Algebra"]].		* The exercise is related to the topic of the chapter [[Channel_Coding/Some_Basics_of_Algebra\| "Some Basics of Algebra"]].
	* Please don't be confused that in the text the set  $ \{a}, { b},{c},{d},{e}\}$  is used and in the tables the set  $ \rm \{a}, { b},{c},{d},{e}\}$. Apologies.		* Please don't be confused that in the text the set  $ \{{a}, { b},{c},{d},{e}\}$  is used and in the tables the set  $ \rm \{{a}, { b},{c},{d},{e}\}$. Apologies.

Noah at 20:30, 26 August 2022

2022-08-26T20:30:47Z

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Guenter at 14:44, 24 August 2022

2022-08-24T14:44:01Z

← Older revision		Revision as of 16:44, 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^]]

Guenter: Guenter moved page Aufgaben:Aufgabe 2.2Z: Galoisfeld GF(5) to Aufgaben:Exercise 2.2Z: Galois Field GF(5)

2022-08-24T14:40:32Z

Guenter moved page Aufgaben:Aufgabe 2.2Z: Galoisfeld GF(5) to Aufgaben:Exercise 2.2Z: Galois Field GF(5)

← Older revision	Revision as of 16:40, 24 August 2022
(No difference)