Question

Please help me with the following proof in Fields(Linear Algebra):
Sow that if we sum all the identity element (1) finite number of times in a finite field we'll get the neutral element (0).

Answer 1

If \( F=\mathbb Z_n \), then if we add the unit element n times, we get
\[ \bar 1 + \bar 1+ \cdots +\bar 1 = \bar n=\bar 0
\]

Answer 2

n is a prime in the above post.

Answer 3

In general read http://mathworld.wolfram.com/FiniteField.html

Answer 4

Identity element means 1*a = a for all a in the ring, and
Neutral element means 0+a = a for all a in the ring, right?

Answer 5

Well, yeah, so... first of all, your finite field is a ring, and therefore an abelian group.
Form the cyclic subgroup generated by 1. It is a subgroup, therefore, it should have the additive identity (0) in it, and being a subgroup of a finite group, it is also finite, say, of order k. And from that, it's done. If <q> is the cyclic subgroup generated by q, and it has order p, then pq=0, where 0 is the additive identity in the group q is associated with.

Answer 6

Every finite filed contains at least one subfield isomorphic to \( \mathbb Z_p\) where p is a prime. So apply my first post.

Answer 7

First of all thanks for the comments but I'm confused. I don't need to prove this under Zn but in general field. I'm trying to understand you last post finding one subfield isomorpic to Zn and I still cant figure it out.
Also what's are P and Q? or you talking about Fermat's Little Theorem or Wilson Theorem. I need a little bit more explanation on this one.

Answer 8

@eliassaab and @terenzreignz

Answer 9

Let F be finite field. Then there is a prime number p and an integer n>0 such that F has p^n elements. Such a field contains the subfield \(\mathbb Z_p\). Hence use my first post.
See http://en.wikipedia.org/wiki/Finite_field_arithmetic

Answer 10

But your proof is only valid over certain values of p and n.
I think the proof should be general, for any field in the world, with any n should apply this property.

Answer 11

@eliassaab

Answer 12

My teachter told me I can solve it using the "Pigeonhole principle".
Because if I look at the following n items :
1, 1+1, 1+1+1 , 1+1+1....+1 (n+1 times)

I have n items and I sum the unity element n+1 times so I must have an element repeats itself and I can conclude from this fact that Its equal to the zero element.

I write all this but I don't understand the conculsion and what equal to zero?
and why do we sum them this way.

Can you explain me this?