Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a and p are relatively prime, we find that there must exist integers x and y such that xa + yp = 1. So?)

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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
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 2.58.3
Now let [ap be a nonzero element in Z/pZ. Show that [a, is
invertible. (Hint: Invoking the fact that a and p are relatively
prime, we find that there must exist integers x and y such that
xa + yp = 1. So?)
Transcribed Image Text:Exercise 2.58.3 Now let [ap be a nonzero element in Z/pZ. Show that [a, is invertible. (Hint: Invoking the fact that a and p are relatively prime, we find that there must exist integers x and y such that xa + yp = 1. So?)
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