1. ## Discrete Quadratic Fields and Integers (Easy, Yes, No Examples)

Hello All,

I had a few finite examples from my text I had some questions about. They are discussing Q[Sqrt(-5)] , and I know that it is i*Sqrt(5) = Sqrt(-5). Given this information, it gives some examples and it said I should be able to identify which examples are in Q[Sqrt(-5)]

Example 1: 2+3*Sqrt(-5)

Example 2: (3+8*Sqrt(-5))/2

Example 3: (3+8*Sqrt(-5))/5

Example 4: 3/5

Example 5: i*Sqrt(-5)

From observation I inferred that Example 5 was in Q[Sqrt(-5)], but then i noticed that it says i*Sqrt(-5) and not i*Sqrt(5). I don't understand how Example 4 could be part of a negative rooted Quadratic field either.

Could someone explain these? If you could, please provide a small reason as to why each one is or isn't part of Q[Sqrt(-5)].

Thank you, all the help is appreciated!

2. Originally Posted by Samson
Hello All,

I had a few finite examples from my text I had some questions about. They are discussing Q[Sqrt(-5)] , and I know that it is i*Sqrt(5) = Sqrt(-5). Given this information, it gives some examples and it said I should be able to identify which examples are in Q[Sqrt(-5)]

Example 1: 2+3*Sqrt(-5)

Example 2: (3+8*Sqrt(-5))/2

Example 3: (3+8*Sqrt(-5))/5

Example 4: 3/5

Example 5: i*Sqrt(-5)

From observation I inferred that Example 5 was in Q[Sqrt(-5)], but then i noticed that it says i*Sqrt(-5) and not i*Sqrt(5). I don't understand how Example 4 could be part of a negative rooted Quadratic field either.

Could someone explain these? If you could, please provide a small reason as to why each one is or isn't part of Q[Sqrt(-5)].

Thank you, all the help is appreciated!
So $a + bi\sqrt{5} \in \mathbb{Q}(\sqrt{-5})$ if and only if $a,b\in\mathbb{Q}$. Example 1 already matches the form, so you just look at it and say "yes". For example 2 you can do simple manipulation to get it into the form (3/2)+(8/2)*Sqrt(-5) which matches the form $a + bi\sqrt{5}$. Likewise for example 3. Example 4, let b = 0, which is acceptable. Example 5 is slightly trickier, just expand to get i*sqrt(-5) = i*i*sqrt(5) = -sqrt(5) which you should be able to see is not in Q[sqrt(-5)].

Edit: Well I interchanged $a + bi\sqrt{5}$ with $a + b\sqrt{-5}$ without meaning to, but it should still be clear enough.

3. Originally Posted by undefined
So $a + bi\sqrt{5} \in \mathbb{Q}[-5]$ if and only if $a,b\in\mathbb{Q}$. Example 1 already matches the form, so you just look at it and say "yes". For example 2 you can do simple manipulation to get it into the form (3/2)+(8/2)*Sqrt(-5) which matches the form $a + bi\sqrt{5}$. Likewise for example 3. Example 4, let b = 0, which is acceptable. Example 5 is slightly trickier, just expand to get i*sqrt(-5) = i*i*sqrt(5) = -sqrt(5) which you should be able to see is not in Q[-5].
Thank you! A quick clarification, so when we have (3/2)+(8/2)*Sqrt(-5), it doesn't matter that a and b are not integers does it? And how do we check if they exist in Q[-5] ?

4. Originally Posted by Samson
Thank you! A quick clarification, so when we have (3/2)+(8/2)*Sqrt(-5), it doesn't matter that a and b are not integers does it? And how do we check if they exist in Q[-5] ?
Notice the "Q" in Q[sqrt(-5)]. So you just check the definition, a+b*Sqrt(-5) where a and b are rational. There is nothing further to check.

5. Originally Posted by undefined
Notice the "Q" in Q[-5]. So you just check the definition, a+b*Sqrt(-5) where a and b are rational. There is nothing further to check.
Okay! Thank you very much for your help! Much props to be awarded!