ℤ and ℤ[x] isomorphism

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In summary, the conversation discusses the existence of a ring isomorphism between the set of integers and the set of polynomials with integer coefficients. The participants consider different arguments and eventually conclude that such an isomorphism cannot exist. They also explore the implications of this conclusion and question the possibility of an injective map from the set of polynomials to the set of integers.
  • #1
Bachelier
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There exists none. What's the easiest way to prove this?
Can we state that all elements of ℤ are in ℤ[x] but not the other way around?
 
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  • #2
Just because you came up with a bad attempt at an isomorphism (the obvious inclusion of the integers) doesn't mean a better one doesn't exist.
 
  • #3
Bachelier said:
There exists none. What's the easiest way to prove this?
Can we state that all elements of ℤ are in ℤ[x] but not the other way around?

I think a ring isomorphism must map 1 to 1 in Z[x] so Z must be mapped to Z in Z[x].

h(m) = h(1.m) = h(1)h(m). So h(1) = 1. I think this is right.
 
  • #4
if there WAS such an isomorphism, we would also have one from Z[x] to Z, say φ.

now 1 ( = 1 + 0x + 0x^2 +...) in Z[x] is a multiplicative identity,

so say φ(f(x)) = k.

so k = φ(f(x)) = φ(1f(x)) = φ(1)φ(f(x)) = φ(1)k.

so φ(1) = 1.

by induction, for n ≥ 1, φ(n) = φ(1+1+...+1) (n times)

= φ(1) + φ(1) + ...+ φ(1) (n times)

= 1 + 1 +...+ 1 (n times)

= n.

similarly, φ(-n) = -φ(n) = -n.

given this, and the fact that we must assign an integer value to φ(x),

how can φ be injective?
 
  • #5


As a scientist, it is important to approach this question with a clear understanding of mathematical definitions and principles. In this case, the concept of isomorphism is crucial.

An isomorphism is a bijective function between two mathematical structures that preserves their algebraic properties. In other words, if two structures are isomorphic, they are essentially the same in terms of their algebraic properties.

In the case of ℤ and ℤ[x], they are both algebraic structures but they have different properties. ℤ is a set of integers, while ℤ[x] is a set of polynomials with integer coefficients. These two structures have different algebraic operations and thus are not isomorphic.

To prove this, we can use the definition of isomorphism and show that there is no bijective function between ℤ and ℤ[x] that preserves their algebraic properties. This can be done by considering the fact that the set of integers does not have an element x, while the set of polynomials does. Therefore, there cannot be a bijective function between the two sets.

It is not accurate to state that all elements of ℤ are in ℤ[x] but not the other way around. This is because the elements of ℤ[x] are polynomials with integer coefficients, while the elements of ℤ are integers. While all elements of ℤ can be represented as polynomials in ℤ[x], not all elements of ℤ[x] can be represented as integers in ℤ.

In conclusion, the isomorphism between ℤ and ℤ[x] does not exist due to their different algebraic properties. This can be proven by considering the definition of isomorphism and the fundamental differences between the two structures.
 

1. What is an isomorphism between ℤ and ℤ[x]?

An isomorphism between ℤ and ℤ[x] is a bijective function that preserves the algebraic structure of the two sets. This means that the function maps integers to polynomials in such a way that the operations of addition, subtraction, and multiplication are preserved.

2. How do you prove that ℤ and ℤ[x] are isomorphic?

To prove that ℤ and ℤ[x] are isomorphic, you need to show that there exists a bijective function between the two sets that preserves the algebraic structure. This can be done by defining a function that maps integers to polynomials in a way that satisfies the definition of an isomorphism.

3. What is the significance of an isomorphism between ℤ and ℤ[x]?

An isomorphism between ℤ and ℤ[x] is significant because it allows us to use the familiar algebraic properties of integers to understand and manipulate polynomials. This can be particularly useful in solving equations involving polynomials.

4. Can any other sets be isomorphic to ℤ and ℤ[x]?

Yes, other sets can be isomorphic to ℤ and ℤ[x]. Isomorphism is a concept that can be applied to any two sets that have a similar algebraic structure. For example, the sets of real numbers and complex numbers are also isomorphic.

5. How does an isomorphism between ℤ and ℤ[x] relate to number theory?

An isomorphism between ℤ and ℤ[x] has implications in number theory as it allows us to view polynomials as a generalization of integers. This can lead to insights and connections between different concepts in number theory, such as prime numbers, factorization, and divisibility.

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