Inner Product Space over Fnite Field

[jOc3]

I come over this in my coding theory but can't understand it. It says finite fields do not fulfil the definition of inner product space like other fields (R and C. Why? How is the proof? Thanks!

 

[learningphysics]

I come over this in my coding theory but can't understand it. It says finite fields do not fulfil the definition of inner product space like other fields (R and C. Why? How is the proof? Thanks!

I think it's just part of the definition. An inner product space is a vector space over R or C, by definition.

 

[HallsofIvy]

I don't think that quite answers the question. Is it true that, if V is a vector space over a finite field, then there cannot exist an function VxV-> V satisfying the requirements for an inner product?

 

[learningphysics]

I don't think that quite answers the question. Is it true that, if V is a vector space over a finite field, then there cannot exist an function VxV-> V satisfying the requirements for an inner product?

Yes, you're right. This wikipedia page on inner product spaces explains it well:

http://en.wikipedia.org/wiki/Inner_product_space#Definition

The explanation is in the remark section of the definition...

One of the requirements of inner product spaces is that <x,x> is nonnegative. But positivity and negativity don't make sense in a finite field.

 

[mathwonk]

of course this is just language. various dot products certainly make sense over any field, just not non negative ones.

 

[jOc3]

This is what I get from wikipedia but I can't figure it out. "...it is necessary to restrict the basefield to R & C in the definition of inner product space. Briefly, the basefield has to contain an ordered subfield (in order for non-negativity to make sense) & therefore has to have characteristic equal to 0. This immediately excludes finite fields." How is it "immediately excludes finite field"? Best if an example can be provided along with the explanation. Thanks a lot!

 

[learningphysics]

This is what I get from wikipedia but I can't figure it out. "...it is necessary to restrict the basefield to R & C in the definition of inner product space. Briefly, the basefield has to contain an ordered subfield (in order for non-negativity to make sense) & therefore has to have characteristic equal to 0. This immediately excludes finite fields." How is it "immediately excludes finite field"? Best if an example can be provided along with the explanation. Thanks a lot!

If you have a non-zero element x in a field... And you add it to itself a number of times > 0... for example x is 1 time... x + x is 2 times... then the minimum number of times needed for the sum to be zero is the characteristic of the field... so given x is non zero, x+x is non-zero, but x + x + x is zero... then the characteristic of the field is 3. If there is no such number greater than zero, then the characteristic of the field is defined to be 0.

So suppose a field has non-zero characteristic. Let x be any non-zero element... suppose it's positive... then x + x + x +... = 0 at some point... but this doesn't make sense... for positivity and negativity to make sense... if you have a positive element, and you keep adding positive elements, the result should be positive... not zero.

Take the field Z₃ = {[0]₃,[1]₃,[2]₃}
[1]₃+[1]₃+[1]₃ = 0. If [1]₃ was positive then the sum would be positive... if [1]₃ was negative then the sum would be negative. positive and negative don't make sense here.

In the real or complex number field 1+1+1+... will never be equal to zero. So they have characteristic 0.

 

[jOc3]

Wow! That really answers all my doubt. Thank you!