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Let $p$ and $q$ be two distinct primes. Pick the correct statements from the following:

a. $Q(\sqrt p)$ is isomorphic to $Q(\sqrt q)$ as fields.

b. $Q(\sqrt p)$ is isomorphic to $Q(\sqrt{−q})$ as vector spaces over Q.

c. $[Q(\sqrt p,\sqrt q) : Q] = 4$.

d. $Q(\sqrt p,\sqrt q) = Q(\sqrt p + \sqrt q)$.

(c) & (d) are correct. (a) is not correct for fields but correct for vector spaces. (b) not sure. i think it is correct by the same arguement of (a).am i right?

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    Is this a homework question, or just one for personal development, as this may effect the type of response that people give?2012-09-17
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    @DavidWard I don't know why it should affect the type of response. If one should not give a full answer to a homework question, many good questions might have no full answer in this site. I think this is against the policy of this site.2012-09-17
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    @MakotoKato My reasoning is that if it is a homework question, then the answer itself is of importance to the person asking the question. However, if the question is not homework, then for a question as above, I would feel happy to give the answer, but then lead someone through the reasoning on how to get there.2012-09-18
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    @DavidWard I'm sorry that I don't understand your reasoning. Could you explain why you don't want to give the answer to a homework question?2012-09-18
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    @MakotoKato My reasoning is as follows: Firstly suppose that the question asked is from an assessed piece of work. Then as the question is a multiple-choice question, the actual answer will attract some if not all of the marks (depending on the desire of the assessor). Thus knowledge of the answer is of critical importance. On the other hand, if the question asked is purely for personal development, whilst leading the member towards the solution, it can be beneficial to actually know where you are being led. It just gives some motivation for where the reasoning is heading.2012-09-18
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    @DavidWard Firstly homework should not be means of assessment(he/she can ask someone about the answer). Secondly, as I wrote, if we decide not to give the answer to a homework question, many good questions will remain unanswered(this will be against the policy of this site).2012-09-18
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    @MakotoKato I never said that I wouldn't give the answer to a homework question, rather the time at which the answer is revealed may differ. However, I think it is best if we just agree to disagree on this point. Math StackExchange is a great site for people to offer one another help let's just agree on that.2012-09-19
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    @DavidWard I just don't understand why you treat homework questions differently from non-homework ones. If you think giving an answer to a homework question soon is pedagogically harmful, the same can be said for a non-homework question. IMO, some(or many) people don't want to give away an answer to a homework question just because they feel like being used, not because they are particularly interested in pedagogy. While I understand their sentiment, it may have negative effects in this site(e.g. only obscure answers are given).2012-09-19

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