Is Q[x]/I ring-isomorphic to Q[\sqrt{2}]?

I can map f(x) + I to a + b\sqrt{2} by f(x) + I \mapsto a + b\sqrt{2}.In summary, to prove that Q[x]/\langle x^2 - 2 \rangle is ring-isomorphic to Q[\sqrt{2}] = \{a + b\sqrt{2} \mid a,b \in Q\}, we can map elements in Q[x]/\langle x^2 - 2 \rangle to elements in Q[\sqrt{2}] by rewriting elements in Q[x] as (xx - 2)q(x) + r(x) and then mapping f(x) + I to a + b\sqrt{2},
  • #1
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Homework Statement
Prove that [tex]Q[x]/\langle x^2 - 2 \rangle[/tex] is ring-isomorphic to [tex]Q[\sqrt{2}] = \{a + b\sqrt{2} \mid a,b \in Q\}[/tex].


The attempt at a solution
Denote [tex]\langle x^2 - 2 \rangle[/tex] by I. [tex]a_0 + a_1x + \cdots + a_nx^n + I[/tex] belongs to Q[x]/I. It has n + 1 coefficients which somehow map to a and b. I don't think any injection can do this. I'm stumped. Any hints?
 
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  • #2
I think I got it: Let f(x) be an element of Q[x]. f(x) may be rewritten as (xx - 2)q(x) + r(x) for some q(x), r(x) in Q[x] with r(x) = 0 or deg r(x) < deg (xx - 2) = 2. Thus, r(x) has the form a + bx, a and b both belonging to Q. Aha!
 

Related to Is Q[x]/I ring-isomorphic to Q[\sqrt{2}]?

1. What is a ring isomorphism problem?

A ring isomorphism problem is a mathematical problem in which one is asked to determine whether two given rings are isomorphic to each other. Isomorphism refers to a mapping between two mathematical structures that preserves their algebraic properties.

2. How do you prove that two rings are isomorphic?

To prove that two rings are isomorphic, one must show that there exists a bijective mapping between the two rings that preserves their algebraic properties. This can be done by showing that the two rings have the same number of elements, the same operations, and the same structure.

3. What is the significance of solving a ring isomorphism problem?

Solving a ring isomorphism problem has practical applications in fields such as cryptography, computer science, and physics. It also helps in understanding the structure and properties of different mathematical objects, leading to further developments in the field of abstract algebra.

4. Are there any techniques for solving a ring isomorphism problem?

There are various techniques for solving a ring isomorphism problem, such as using homomorphisms, finding generators and relations, and using invariants. The choice of technique depends on the specific problem and the properties of the given rings.

5. Can a ring isomorphism problem have more than one solution?

Yes, a ring isomorphism problem can have more than one solution. This happens when there are multiple mappings between the two rings that preserve their algebraic properties. In such cases, one can choose the most suitable or simplest solution based on the context of the problem.

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