Exercise 2.58.2 Show that the zero in this ring is [0p, and the 1 in this ring is [1]p. (In particular, a, is nonzero in Z/pZ precisely when a is not divisible by p.)
Exercise 2.58.2 Show that the zero in this ring is [0p, and the 1 in this ring is [1]p. (In particular, a, is nonzero in Z/pZ precisely when a is not divisible by p.)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![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%2Fc65c3cbf-447e-4b51-a56b-c402c5c4a301%2F5a6rmf_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
![Exercise 2.58.2
Show that the zero in this ring is 0p, and the 1 in this ring is
[1]p. (In particular, [a, is nonzero in Z/pZ precisely when a is not
divisible by p.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2Fc65c3cbf-447e-4b51-a56b-c402c5c4a301%2Fz1arlo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 2.58.2
Show that the zero in this ring is 0p, and the 1 in this ring is
[1]p. (In particular, [a, is nonzero in Z/pZ precisely when a is not
divisible by p.)
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