Show that addition and multiplication are well-defined, that is, if a~ a' and b~ b', then a +b~ a' +b and a b~d.b'.
Show that addition and multiplication are well-defined, that is, if a~ a' and b~ b', then a +b~ a' +b and a b~d.b'.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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abstract algebra
![Example 2.58
However, Examples 2.55 and 2.56 do generalize suitably: it turns out
that for any prime p, the ring Z/pZ is a field (with p elements). Recall
from the discussions in Examples 2.20 and 2.21 that the elements of
Z/pZ are equivalence classes of integers under the relation a ~bif and
only if a-b is divisible by p. The equivalence class [a], of an integer a is
thus the set of integers of the form a + p, a ± 2p, a ±3p, .... Addition
and multiplication in Z/pZ are defined by the rules
1. [a]p + [b, = [a + blp
2. [a), (b), = [a - b)p](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2Fc42326e1-0f67-4f16-8ce0-d08220e08eb3%2F416r5b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example 2.58
However, Examples 2.55 and 2.56 do generalize suitably: it turns out
that for any prime p, the ring Z/pZ is a field (with p elements). Recall
from the discussions in Examples 2.20 and 2.21 that the elements of
Z/pZ are equivalence classes of integers under the relation a ~bif and
only if a-b is divisible by p. The equivalence class [a], of an integer a is
thus the set of integers of the form a + p, a ± 2p, a ±3p, .... Addition
and multiplication in Z/pZ are defined by the rules
1. [a]p + [b, = [a + blp
2. [a), (b), = [a - b)p
Transcribed Image Text:you
the
Exercise 2.58.1
yours
Show that addition and multiplication are well-defined, that is, if
a~a' and b ~b', then a +b~d+b and a ·b~d.b.
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