Question

In: Advanced Math

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F...

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F .

Show further that the smallest positive integer with this property is a prime number.

Expert Solution

Given data,

Let F be a finite field

Let char(F) = m

Then

,

In particular,

Hence,

=

upto m times

Therefore, the asked n is of the form n = mK, .

Let char(F) =

If possible suppose that be not prime

Let = rs,(a conjugate number where r,s > 1).

So, .a = (by definition of characteristic)

In particular, =

=> =

=>

=> or

(as F is a field, hence it is a integral domain)

char(F) = r < or char(F) = S <

- a contradiction.

(as is the least positive integer such that = )

Hence, char(F) = p, a prime.

Colby Messinger answered 2 years ago

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