What if the (semi) field characteristic of N is not zero?

In summary, the characteristic of a field is defined as the minimum number of times the multiplicative identity element must be used in a sum to get the additive identity element. This rule can also be applied to the set of natural numbers. However, there is a problem when using the Riemann zeta function to calculate the sum of an infinite number of identity elements. If we group the numbers differently, the sum will yield a different value. Additionally, attempting to do the sum a finite number of times and then adding one more identity element does not produce a meaningful result. The reason for this lies in the fact that changing the grouping of numbers changes the way their sum is calculated. This is not a conventional method and should not be used.
  • #1
Cathr
67
3
By definition, the characteristic of a field is the smallest number of times one must use the ring's multiplicative identity element (1) in a sum to get the additive identity element (0). Can we use the same rule for the set of natural numbers?

If yes, I found a problem, that has something to do with the Riemann zeta function.

If we calculate the sum for an infinity of identity elements, we obtain the sum of all natural numbers: 1+(1+1)+(1+1+1)+(1+1+1+1)+..., and, if we calculate this sum using the Riemann zeta function, we obtain
242b35184f3c25d8af6a1fa0e992a015ca8b07f8
.

Do it 12 times and add one more multiplicative identity - what we obtain is exactly zero.

Does this make sense? Or it is wrong right from the beginning?
 
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  • #2
Cathr said:
If we calculate the sum for an infinity of identity elements, we obtain the sum of all natural numbers: 1+(1+1)+(1+1+1)+(1+1+1+1)+...,
Why do you group them like that?
There are methods to assign a finite value to divergent series (e. g. Ramanujan summation), but that value changes if you introduce arbitrary groups like that.
Group it as 1+4+9+16+25+... and the Ramanujan sum is 0.
Cathr said:
Do it 12 times and add one more multiplicative identity
You cannot "do an infinite sum 12 times in a row and then plus one" in a meaningful way.
 
  • #3
I never really understood why changing the way of grooping numbers changes their sum. For me they just have different intervals after which we visibly insert a "+", but they all contain natural numbers that go to infinity. Or is it a convention, which means that we look at each number as a term? Does it mean that 1+4+9+16+25... is "denser" than 1+2+3+4+5... and therefore it has a bigger partial sum?
 
  • #4
Cathr said:
I never really understood why changing the way of grooping numbers changes their sum.
It is not a proper sum.
Forget Ramanujan summation. It is an exotic method to assign a number to divergent series that you'll never use.
 

1. What is a (semi) field characteristic of N?

The (semi) field characteristic of N refers to the smallest positive integer n such that n multiplied by any element in the field will result in 0. In other words, it represents the number of times an element needs to be added to itself to equal 0 in the field.

2. What happens if the (semi) field characteristic of N is not zero?

If the (semi) field characteristic of N is not zero, then it means that there exists an element in the field that, when multiplied by n, will not result in 0. This element is known as the characteristic element and it determines the structure and properties of the field.

3. Can the (semi) field characteristic of N be a negative number?

No, the (semi) field characteristic of N must always be a positive integer. This is because the characteristic element must be a non-zero element in the field, and a negative number multiplied by a positive integer will always result in a negative number.

4. How does the (semi) field characteristic of N affect mathematical operations in the field?

The (semi) field characteristic of N plays a crucial role in determining the behavior of mathematical operations in the field. For example, in a field with characteristic 3, adding an element to itself three times will result in 0. This property is used in many algebraic equations and proofs.

5. What are some real-life applications of understanding the (semi) field characteristic of N?

Understanding the (semi) field characteristic of N is essential in many areas of mathematics, such as abstract algebra and number theory. It also has practical applications in fields such as coding theory, cryptography, and computer science. In coding theory, the (semi) field characteristic of N is used to construct error-correcting codes, while in cryptography, it is used in the creation of secure encryption algorithms.

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