ring isomorphism proof,need help for my homework

**canuhelp** · May 9th 2009, 07:46 AM

How can i prove that $Q[_{\sqrt{2}}]$ is a ring isomorphism to $Q[_{\sqrt{3}}]$

**TheAbstractionist** · May 9th 2009, 08:20 AM

Originally Posted by canuhelp

How can i prove that $Q[_{\sqrt{2}}]$ is a ring isomorphism to $Q[_{\sqrt{3}}]$

Hi canuhelp.

Try the mapping $f\mathbb Q(\sqrt2)\to\mathbb Q(\sqrt3),$ $f\left(a+b\sqrt2\right)\ \to\ a+b\sqrt3.$

**ThePerfectHacker** · May 9th 2009, 10:25 AM

Originally Posted by canuhelp

How can i prove that $Q[_{\sqrt{2}}]$ is a ring isomorphism to $Q[_{\sqrt{3}}]$

Originally Posted by TheAbstractionist

Hi canuhelp.

Try the mapping $f\mathbb Q(\sqrt2)\to\mathbb Q(\sqrt3),$ $f\left(a+b\sqrt2\right)\ \to\ a+b\sqrt3.$

I disagree with TheAbstractionist because $\mathbb{Q}(\sqrt{2})$ is not isomorphic to $\mathbb{Q}(\sqrt{3})$ as fields (so they cannot be isomorphic as fields). Now since $\sqrt{2},\sqrt{3}$ are algebraic integers it means $\mathbb{Q}[\sqrt{2}] = \mathbb{Q}(\sqrt{2}),\mathbb{Q}[\sqrt{3}] = \mathbb{Q}(\sqrt{3})$ . Thus, asking for a ring isomorphism between $\mathbb{Q}[\sqrt{2}]$ and $\mathbb{Q}[\sqrt{3}]$ is asking for an isomorphism between $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ which does not exist.

(They happen to be isomorphic as vector spaces, maybe that is what the poster was really asking).

**vemrygh** · May 9th 2009, 01:42 PM

Construct a map $\phi:\mathbb{Q}[\sqrt{2}]\rightarrow\mathbb{Q}[\sqrt{3}]$ such that: $\phi(\sqrt{2})=\sqrt{3}$ and $\phi(q)=q$ , for all $q\in\mathbb{Q}$ .
Then check that $\phi$ is bijective and is a homomorphism, to show that $\phi$ is an isomorphism.

**TheAbstractionist** · May 10th 2009, 01:12 AM

Hi ThePerfectHacker.

Thanks for pointing this out. I wasn’t thinking clearly when I made my post.

Originally Posted by ThePerfectHacker

(They happen to be isomorphic as vector spaces, maybe that is what the poster was really asking).

That was probably what I had in mind.

**sah_mat** · May 10th 2009, 04:26 AM

i tried but i am not sure it is true i wish somebody will check the solution.

Let $\phi:Q[\sqrt{2}]->Q[\sqrt{3}]$ be a supposed ring isomorphism. Since $\phi(x)=\phi(1*x)=\phi(1)*\phi(x)$ and $\phi$ is onto, we must have $\phi(1)=1.$ This implies that $\phi(2)=2.$ Let $\phi(\sqrt{2})=a+b*\sqrt{3}$ , so 2= $\phi(2)=\phi(\sqrt{2})^2=a^2+3*b^2+2*a*b*\sqrt{3}$ . Since the $\sqrt{3}$ is irrational we must have either a=0 or b=0. Then we use either $\sqrt{2}$ or $\sqrt{2/3}$ irrational to conclude it is impossible to solve this equation with rational numbers.

**Isomorphism** · May 10th 2009, 10:55 AM

Originally Posted by canuhelp

hi thanks for your helps,i asked the same question under advanced topics
and a friend give an answer what do you think about it
http://www.mathhelpforum.com/math-he...-homework.html

Originally Posted by vemrygh

Construct a map $\phi:\mathbb{Q}[\sqrt{2}]\rightarrow\mathbb{Q}[\sqrt{3}]$ such that: $\phi(\sqrt{2})=\sqrt{3}$ and $\phi(q)=q$ , for all $q\in\mathbb{Q}$ .
Then check that $\phi$ is bijective and is a homomorphism, to show that $\phi$ is an isomorphism.

By the suggestion above:

$\phi(2) = 2$ and $\phi(2) = (\phi(\sqrt{2}))^2 = 3$

Thus the suggestion is not well defined!!

canuhelp, did you read post #3?

Also, I dont think its a good idea to post the same question in multiple places. It is against forum rules.

**sah_mat** · May 10th 2009, 11:41 AM

dear Isomorphism canyou give us a complete explanation about this question as thinking this question to find a RİNG isomorphism beetween
$Q[\sqrt{2}]->Q[\sqrt{3}]$

**Swlabr** · May 14th 2009, 02:23 AM

Originally Posted by vemrygh

Construct a map $\phi:\mathbb{Q}[\sqrt{2}]\rightarrow\mathbb{Q}[\sqrt{3}]$ such that: $\phi(\sqrt{2})=\sqrt{3}$ and $\phi(q)=q$ , for all $q\in\mathbb{Q}$ .
Then check that $\phi$ is bijective and is a homomorphism, to show that $\phi$ is an isomorphism.

That doesn't work as then $2$ maps to $3$ and so $1$ maps to $3/2$ , which is absurd:

$\sqrt{3} = (\sqrt{2})\phi = (\sqrt{2}) \phi*(1) \phi = \sqrt{3}*3/2$ . Thus $3/2 = 1$ .

**Swlabr** · May 14th 2009, 05:56 AM

Originally Posted by canuhelp

How can i prove that $Q[_{\sqrt{2}}]$ is a ring isomorphism to $Q[_{\sqrt{3}}]$

What does the question ask you? "Give an isomorphism..." or "Is there an isomorphism..."?

I suspect it is the latter; the solution boils down to the question of "is $\sqrt{3/2}$ a rational number", which i don't know off the top of my head.

Hint: Assume there exists an isomorphism. Some element maps to $\sqrt{3}$ - what must it look like?

EDIT: I am entirely confused - I thought I was only the second person to post in this thread, after vemrygh. Thus my repetition of other peoples posts. I'm sure I read the entire thread first though!

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