Discrete Quadratic Fields and Integers (Easy, Yes, No Examples)
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!