Does there exist a canonical projection from Z^p-1 to Z_p

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In summary, there is a "natural" projection from Zp-1 to Z/p, obtained by looking at the coproduct of the p-1 natural maps Z to Z/p. The p-adic integers can be constructed by taking the inverse limit of the sequence Z/p3 to Z/p2 to Z/p, or by completing Q with respect to the p-adic norm and considering all elements with a p-adic norm less than or equal to one.
  • #1
NoodleDurh
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Okay, I thought that there would exist a projection from ##\mathbb{Z}^{p-1}## to ##\mathbb{Z}_{p}## where ##p## is a prime. Like there exist a projection from ##\mathbb{Z}## to ##\mathbb{Z}_p##
 
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  • #2
Assuming Zp denotes the integers modulo p (and not the p-adic integers), then there is a "natural" projection Zp-1Z/p obtained simply by looking at the coproduct of the p-1 natural maps ZZ/p.
 
  • #3
okay, yeah you assumption is correct. Also, not to get to far off topic, but what does the p-adic integers look like.
 
  • #4
NoodleDurh said:
Also, not to get to far off topic, but what does the p-adic integers look like.

Just consider the inverse limit of the sequence ...→Z/p3Z/p2Z/p and this produces the p-adic integers. Another way is completing Q with respect to the p-adic norm and then consider all element with p-adic norm less than or equal to one.
 
  • #5
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Yes, there does exist a canonical projection from ##\mathbb{Z}^{p-1}## to ##\mathbb{Z}_{p}## where ##p## is a prime. This projection is defined by taking the elements of ##\mathbb{Z}^{p-1}## and reducing them modulo ##p##, resulting in elements in the finite field ##\mathbb{Z}_{p}##. This is similar to how there exists a projection from ##\mathbb{Z}## to ##\mathbb{Z}_{p}## by reducing elements modulo ##p##. The canonical projection from ##\mathbb{Z}^{p-1}## to ##\mathbb{Z}_{p}## is useful in various mathematical and scientific contexts, such as in number theory and algebraic geometry.
 

Related to Does there exist a canonical projection from Z^p-1 to Z_p

1. What is a canonical projection?

A canonical projection is a mathematical function that maps elements of one set to corresponding elements of another set in a natural, "canonical" way. In this case, we are considering the sets Z^(p-1) and Z_p, which represent the integers modulo p-1 and p, respectively.

2. Why is a canonical projection important in the context of Z^(p-1) and Z_p?

A canonical projection is important because it allows us to create a one-to-one correspondence between elements of Z^(p-1) and Z_p, simplifying computations and allowing for easier analysis of these sets.

3. Does a canonical projection always exist between two sets?

No, a canonical projection does not always exist between two sets. It depends on the specific sets and their properties. In the case of Z^(p-1) and Z_p, a canonical projection does exist.

4. What is the significance of p-1 in this context?

The p-1 in Z^(p-1) represents the group of integers that are relatively prime to p, and is important because it allows us to create a finite set with the same number of elements as Z_p, making a canonical projection between the two sets possible.

5. Are there any practical applications of a canonical projection from Z^(p-1) to Z_p?

Yes, there are several practical applications of this canonical projection. One example is in cryptography, where it is used in algorithms involving modular arithmetic. It is also used in number theory and other branches of mathematics.

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