Jacobi Sums Explained: A Simple Guide with Examples

[johnkubik]

Would someone be kind enough to explain Jacobi sums in a simple manner using actual numbers. I have read over the math jingo 100 times and have no clue what it actually does.

Thanks!

Edit: Here is a link to the wiki of the Jacobi sums. http://en.wikipedia.org/wiki/Jacobi_sum

 

[DonAntonio]

Would someone be kind enough to explain Jacobi sums in a simple manner using actual numbers. I have read over the math jingo 100 times and have no clue what it actually does.

Thanks!

Edit: Here is a link to the wiki of the Jacobi sums. http://en.wikipedia.org/wiki/Jacobi_sum

For you to actually understand basically this stuff it is important you've already taken a basic course in

group theory and some basic analytic number theory. If this is so, then you should be able to point exactly at where

you're having problems understanding these things, and not ask in general for an explanation.

DonAntonio

 

[johnkubik]

Yeah, not really sure where to even start. Would you be kind enough to just run through an example of it with some actual numbers?

Thanks!

 

[DonAntonio]

Yeah, not really sure where to even start. Would you be kind enough to just run through an example of it with some actual numbers?

Thanks!

No unless you understand how are Dirichlet characters defined and for that you need a basic grasp of group theory...and then you

alone can write down lots of numerical examples!

DonAntonio

 

[CellCoree]

Sure, I can try to explain Jacobi sums in a simpler way using actual numbers.

Jacobi sums are a mathematical concept that is used in number theory and modular arithmetic. They were first introduced by the mathematician Carl Gustav Jacobi in the 19th century.

To understand Jacobi sums, it's helpful to first understand modular arithmetic. In modular arithmetic, we are working with numbers that are "mod" a certain number. For example, in mod 5 arithmetic, we only work with numbers that are multiples of 5 (5, 10, 15, etc) and any number that is not a multiple of 5 is considered to be 0. So 7 would be equivalent to 2 in mod 5 arithmetic.

Now, Jacobi sums come into play when we are working with two different moduli (plural of modulus). Let's say we have two moduli, 5 and 7. We can create a table with all the possible combinations of numbers that are "mod" 5 and 7. So our table would have numbers like (0,0), (1,0), (2,0), (3,0), (4,0), (0,1), (1,1), (2,1), (3,1), (4,1), etc.

Jacobi sums are used to calculate the values in this table. Let's say we want to calculate the value for (2,3). We would take the Jacobi symbol of 2 and 3, which is a number that represents the relationship between 2 and 3 in mod 5 arithmetic. In this case, the Jacobi symbol of 2 and 3 would be -1.

Next, we take this Jacobi symbol and raise it to the power of the sum of the two numbers (2+3=5). So (-1)^5 = -1. This -1 is the value of the Jacobi sum for (2,3).

In short, Jacobi sums are used to calculate the values in a table of all possible combinations of numbers that are "mod" two different moduli. They use the Jacobi symbol and raise it to the power of the sum of the two numbers to get the value for each combination.

I hope this helps to clarify Jacobi sums for you. Let me know if you have any other questions.