WildTrig20: Trigonometry with Finite Fields (II)

This video explains more about arithmetic in a finite field, shows how to visualize points in a finite plane, and how to compute quadrance and spread. Examples are taken from the field with seven elements.

Category:

Science & Technology

Tags:

• Finite
• fields
• rational
• trigonometry
• mathematics
• arithmetic
• Fermat's
• theorem
• quadrance
• spread

[mshomsi]

thank you so much:

I wounder the difference between what have been said and the Galois Fields where evry field can be presented as polynomial such as:

GF (2)=x^2 + X+1.

Regards,

[njwildberger]

@mshomsi As per my comment below to JohnCosmas, there are other finite fields besides the prime fields I describe here, and they indeed can be constructed using polynomials.

[JohnCosmas]

Norman, thanks for the great video. It really helps us engineers understand some of the maths behind the technologies. In communications, fields are made from prime numbers as well as prime (primitive) polynomials (representing numbers that are powers of prime numbers e.g. 2^2 and 2^3) in order to obtain this unique mapping of elements that you explained.

Presumably the proof of this unique mapping at time 3.00 minutes into your video must also apply but I can not quite see how. Can you help?

[njwildberger]

Hi JohnCosmas,

The finite fields which I describe are the prime fields, and they are relatively simple. There is one for each prime number. However as you say there are also OTHER finite fields with a prime power of elements, like 9 or 25.

These prime power fields are more indirect and complicated to describe, and use the theory of irreducible polynomials over the prime fields. I will sometime put up some videos that explain this. Thanks for the comment.

[surferblue976]

this is great stuff! im using it for error correction codes and image stegenography

[njwildberger]

Hi @surferblue976,

That sounds pretty interesting. If you write something up, please post it or send it to me. 

[tedtdu]

@surferblue976

@njwildberger

Sure, finite field is mostly used for error correction, but make sure you have very good understand of linear code and matrix

[tedtdu]

This man loves his job very much for sure and the black president should thank him

[brangelito]

I've never seen that proof of Fermats little theorem before, very neat!

[theboombody]

Wow, the first video I've seen on youtube on number theory. I feel like I'm back in my second year of college. It's fun to invent new number systems.