Field of zero characteristics

In summary: Finally, since every nonzero element of P is the product of some number in P with 1, this is also a homomorphism from P to F, and so the theorem is proved.In summary, the theorem states that every field with zero characteristic has a prime subfield isomorphic to ℚ.
  • #1
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I am interested in the following theorem:
Every field of zero characteristics has a prime subfield isomorphic to ℚ.
I am following the usual proof, where we identify every p∈ℚ as a/b , a∈ℤ,beℕ, and define h:ℚ→P as h(a/b)=(a*1)(b*1)-1 (where a*1=1+1+1... a times) I have worked out the multiplicative rule for this homomorphism, but I am not sure how to prove the additive. I am also interested in what does the notation
(a*1)(b*1)-1 exactly mean. Thanks.
 
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  • #2
Danijel said:
I am interested in the following theorem:
Every field of zero characteristics has a prime subfield isomorphic to ℚ.
I am following the usual proof, where we identify every p∈ℚ as a/b , a∈ℤ,beℕ, and define h:ℚ→P as h(a/b)=(a*1)(b*1)-1 (where a*1=1+1+1... a times) I have worked out the multiplicative rule for this homomorphism, but I am not sure how to prove the additive. I am also interested in what does the notation
(a*1)(b*1)-1 exactly mean. Thanks.
All we know about the prime field is its characteristic and ##0,1 \in \mathbb{P}##. ##a\in \mathbb{P}## is not sure. However ##\underbrace{1+\ldots +1}_{a \text{ times}}=a \,\cdot \, 1## for ##a \in \mathbb{N}## is, and likewise for negative ##a##. That's why ##a## times ##1## is used instead of ##a##. We only have isomorphic images of ##a## in ##\mathbb{P}##.

For the homomorphism, I guess ##\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}## will be needed.
 
  • #3
this probably won't help much until you have some commutative algebra but it gives the outline. First of all since the field F is an abelian group, there is exactly one additive homomorphism from the integers Z into F for each choice of the image of 1 from Z, so choose that image to be the unit element 1 in P. Now you have the unique additive homomorphism from Z to P that fresh described sending each positive integer n to 1+...+1 (n times). By definition of the multiplication in Z this is also a multiplicative map hence a ring map. Then since each non zero integer goes to an invertible element of F, (because F has characteristic zero), there is a unique extension of the previously defined ring homomorphism Z-->F to a ring homomorphism Q --> F. Then since Q is a field, this ring map is necessarily injective, hence defines an isomorphism onto the smallest subfield of F, i.e. the prime field.
 

1. What is a "field of zero characteristics"?

A "field of zero characteristics" is a mathematical concept used in abstract algebra. It refers to a field in which the characteristic, or smallest positive integer that when multiplied by any element in the field gives the identity element, is equal to zero. In other words, there exists no nonzero element that when multiplied by the characteristic gives zero.

2. How is a "field of zero characteristics" different from other fields?

Unlike other fields, a "field of zero characteristics" does not have a characteristic that is a prime number. This means that there exists no finite number of elements that can be added to themselves to get the identity element, which is usually the number 1. In addition, a "field of zero characteristics" does not have a finite number of elements.

3. What are some examples of "fields of zero characteristics"?

Some examples of "fields of zero characteristics" include the field of rational numbers, the field of real numbers, and the field of complex numbers. These fields have a characteristic of zero and an infinite number of elements.

4. What are the practical applications of "fields of zero characteristics"?

"Fields of zero characteristics" have many practical applications in mathematics and science. They are used in cryptography, coding theory, and number theory. They also have applications in physics and engineering, such as in the study of vector spaces and linear algebra.

5. How does the concept of "field of zero characteristics" relate to other mathematical concepts?

"Fields of zero characteristics" are closely related to other mathematical concepts such as fields, rings, and groups. They are a special type of field and share many properties with other fields, such as closure, associativity, and distributivity. They also have connections to other mathematical structures, such as vector spaces and modules.

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