2. Proof Problem (Fields): We have seen several examples of fields; e.g. Zp, R, Q, or C. It turns out that all fields have properties similar to those of Z, or the other three examples. Let F be a field. The characteristic Char(F) of F is the smallest non-negative integer n such that na = a + a +...+ a(n-times) = 0. %3D If there is no such n then Char(F)=0. Prove that the characteristic of a field is either 0 or a prime number p. Your proof should only use theorems or properties we have talked about in class.
2. Proof Problem (Fields): We have seen several examples of fields; e.g. Zp, R, Q, or C. It turns out that all fields have properties similar to those of Z, or the other three examples. Let F be a field. The characteristic Char(F) of F is the smallest non-negative integer n such that na = a + a +...+ a(n-times) = 0. %3D If there is no such n then Char(F)=0. Prove that the characteristic of a field is either 0 or a prime number p. Your proof should only use theorems or properties we have talked about in class.
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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Transcribed Image Text:2. Proof Problem (Fields): We have seen several examples of fields; e.g.
Zp, R, Q, or C. It turns out that all fields have properties similar to those
of Z, or the other three examples. Let F be a field. The characteristic
Char(F) of F is the smallest non-negative integer n such that
na = a + a + ...+ a(n-times)
If there is no such n then Char(F)=0. Prove that the characteristic of a
field is either0 or a prime number p. Your proof should only use theorems
or properties we have talked about in class.
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