Solving Isomorphic Rings: \mathbb{F}_5[x]/(x^2+2) & \mathbb{F}_5[x]/(x^2+3)

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In summary, the conversation revolves around proving that \mathbb{F}_5[x]/(x^2+2) and \mathbb{F}_5[x]/(x^2+3) are isomorphic as rings. The method discussed involves finding a linear homomorphism \phi:R\to S such that \phi(1)=1 and determining what x should be mapped to in order for the isomorphism to work. The suggestion is made to try x --> 2x, which satisfies the necessary property. It is also mentioned that finite fields with the same cardinality are isomorphic and using this fact, it can be concluded that both fields have 25 elements. However, it is ultimately decided that
  • #1
JackTheLad
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Hi guys,

I'm trying to show that [tex]\mathbb{F}_5[x]/(x^2+2)[/tex] and [tex]\mathbb{F}_5[x]/(x^2+3)[/tex] are isomorphic as rings.

As I understand it, I have to find the homomorphism [tex]\phi:R\to S[/tex] which is linear and that [tex]\phi(1)=1[/tex].

I'm just struggling to find what I need to send [tex]x[/tex] to in order to get this work.
 
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  • #2
Well, what property must the image of x satisfy?


If all else fails, there aren't many possibilities, you could just try them all.
 
  • #3
Actually, I think x --> 2x might do it, because

[tex]x^2 + 2 \equiv 0[/tex]
[tex](2x)^2 + 2 \equiv 0[/tex]
[tex]4x^2 + 2 \equiv 0[/tex]
[tex]4(x^2 + 3) \equiv 0[/tex]
[tex]x^2 + 3 \equiv 0[/tex]

Is that all that's required?
 
  • #4
Do you have to provide an explicit isomorphism? If not you can just use the fact that finite fields with the same cardinality are isomorphic...both of these fields are generated by adjoining a root of an irreducible quadratic to a field of order 5, and hence both have 25 elements.
 
  • #5
Yeah, unfortunately I do have to show the explicit isomorphism (we're supposed to do it 'the long way')
 

FAQ: Solving Isomorphic Rings: \mathbb{F}_5[x]/(x^2+2) & \mathbb{F}_5[x]/(x^2+3)

What are isomorphic rings?

Isomorphic rings are rings that have the same structure and operations, but may differ in the specific elements and values they contain. In other words, they are essentially the same mathematical object, just represented differently.

How do you determine if two rings are isomorphic?

To determine if two rings are isomorphic, you can check if they have the same number of elements, the same number of operations, and if these operations follow the same rules. Additionally, you can check if there is a bijective function (a one-to-one and onto mapping) between the two rings that preserves the structure and operations.

3. How do you solve for isomorphic rings?

To solve for isomorphic rings, you can use the definition of isomorphism and check if there is a bijective function between the two rings that preserves the structure and operations. You can also use theorems and properties of isomorphism to simplify the process.

4. What is the relationship between isomorphic rings and congruence?

Isomorphic rings and congruence are related in that they both involve a mapping of elements between two rings. In isomorphism, the mapping is bijective and preserves the structure, while in congruence, the mapping is not necessarily bijective and may only preserve certain properties, such as the remainder after division.

5. How does isomorphism apply to the specific example of \mathbb{F}_5[x]/(x^2+2) & \mathbb{F}_5[x]/(x^2+3)?

In this example, both rings are isomorphic to the finite field \mathbb{F}_{25}, but are represented differently. The isomorphism between these two rings can be described by the mapping of elements from \mathbb{F}_{25} to \mathbb{F}_5[x]/(x^2+2) and \mathbb{F}_5[x]/(x^2+3) based on their remainders after division by x^2+2 and x^2+3, respectively.

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