Consider Z7 =[0,1,2,3......6+7,*7], show that Z7 is a field.
Answers
Answer:
proved
Step-by-step explanation:
Given Consider Z7 =[0,1,2,3......6+7,*7], show that Z7 is a field.
We know addition modulo 7. We can write it as
+7 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
3 3 4 5 6 0 1 2
4 4 5 6 0 1 2 3
5 5 6 0 1 2 3 4
6 6 0 1 2 3 4 5
So Z is closed under +7 . Now this modulo is associative. So we can see inverse of 1 is 6.
The additive identity is 0.
Here if a, b belongs to Z7 , a+7 b = b +7 a for all a,b belongs to Z7
Therefore Z7 is an additive group with respect to +7.
Now consider multiplication modulo 7
X7 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 1 3 5
3 0 3 6 2 5 1 4
4 0 4 1 5 2 6 3
5 0 5 3 1 6 4 2
6 0 6 5 4 3 2 1
So Z7 is closed with respect to x7
a, b belongs to Z7 implies a x7 b belongs to Z for all a, b belongs to Z
Now we have a, b, c belongs to Z7,
a x7 (b +7 c) = a x7 b +=7 a x x7c
(a +7 b) x7 c = a x7 c +7b x7 c is true for all a,b,c belongs to Z7
We know that 1 is multiplicative identity of Z7.
Each non-zero element of Z7 has a multiplicative inverse. So the numbers of Z7 are 1,2,3,4,5,6. These elements are prime to 7.
Therefore Z7 is a field.
Answer:
digit 7 multiplication modulo answer