[Number Theory] Finding principal ideals in Z[√-6]

In summary, to find all ideals in Z[√-5] with the element 6, we can use the hint given by the lecturer. This states that an ideal contains 6 if and only if it contains (2)(3), which can be written as (6) in a principal ideal form. Therefore, we need to factor (2) and (3) in Z[√-5]. The general form for the factorization is a = paqbrc, where a is an element of {0, 1, 2} and b, c are elements of {0, 1}. There are 9 possible combinations, but it is not necessary to check all of them as all 9 combinations
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[Number Theory] Find all the ideals with the element 6 in them in Z[√-5]

Edited original question since I have now found the answer (I realize the title is inconsistent on the forum page), instead I am now trying to do part i) here

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Is it possible to it this way:

2hmzyqd.png


Or is the structure of the question meaning my lecturer wants the actual ideals, and not just a general form? If so, how do i find which ideals contain 6.

I know the general form is

a = paqbrc

Whre a is an element of {0,1,2}, and b,c are an element of {0,1}

So there are 9 different combinations I think, do I have to check all 9? Or do all 9 have this property and there is no need to check?

Thanks
 
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  • #2
Hint: An ideal contains 6 iff it contains (6)=(2)(3) (principal ideals) iff it divides (2)(3).

So factor (2) and (3) in Z[sqrt{-6}].
 

FAQ: [Number Theory] Finding principal ideals in Z[√-6]

1. What is a principal ideal in Z[√-6]?

A principal ideal in Z[√-6] is an ideal that is generated by a single element. In other words, it is the set of all multiples of a single element, including 0.

2. How do you find principal ideals in Z[√-6]?

To find principal ideals in Z[√-6], you can use the Euclidean algorithm to find the greatest common divisor of two elements in Z[√-6]. The ideal generated by this GCD will be a principal ideal.

3. What is the difference between a principal ideal and a non-principal ideal in Z[√-6]?

A principal ideal is generated by a single element, while a non-principal ideal is generated by more than one element. Principal ideals are easier to work with and have simpler properties than non-principal ideals.

4. Can Z[√-6] have prime ideals?

Yes, Z[√-6] can have prime ideals. In fact, every principal ideal in Z[√-6] is also a prime ideal. However, there are also non-principal prime ideals in Z[√-6] that are generated by more than one element.

5. How are principal ideals used in number theory?

Principal ideals are used in number theory to study the divisibility and factorization properties of numbers in Z[√-6]. They are also used in algebraic number theory to study the properties of algebraic number fields, such as their class groups and unit groups.

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