Proving Numbers as Primes in the Ring ( mathbb{Z}[sqrt{-5}] )

When dealing with ring theory in mathematics, the concept of prime elements is crucial. In this article, we will explore how to show that the numbers 3, 7, (4 - sqrt{-5}), and (4 sqrt{-5}) are prime within the ring (mathbb{Z}[sqrt{-5}]).

First, we establish a clear definition: an element (p) in a ring (R) is considered prime if:

It is not a unit, meaning it does not have a multiplicative inverse in (R). When (p) divides a product (ab), it must divide at least one of (a) or (b).

Step 1: Checking for Units

Before we check if the given numbers are prime, we must first determine if any of them are units in (mathbb{Z}[sqrt{-5}]).

In this ring, the units are of the form (u a bsqrt{-5}) such that the norm (N(u) a^2 - 5b^2 1). The only units in this ring are (1) and (-1).

For 3:
The norm (N(3) 3^2 9 eq 1), so 3 is not a unit.

For 7:
The norm (N(7) 7^2 49 eq 1), so 7 is not a unit.

For (4 - sqrt{-5}):
The norm (N(4 - sqrt{-5}) 4^2 - 5 cdot 1^2 16 - 5 11 eq 1), so (4 - sqrt{-5}) is not a unit.

For (4 sqrt{-5}):
The norm (N(4 sqrt{-5}) 4^2 - 5 cdot 1^2 16 - 5 11 eq 1), so (4 sqrt{-5}) is not a unit.

Since all the numbers are not units, we can now proceed to the next step.

Step 2: Proving Primality

To confirm that these elements are prime in (mathbb{Z}[sqrt{-5}]), we need to apply the second criterion: if (p) divides (ab), then (p) must divide at least one of (a) or (b).

For 3 and 7:

Both 3 and 7 are prime numbers in (mathbb{Z}), and thus they remain prime in (mathbb{Z}[sqrt{-5}]). If 3 divides (ab) in (mathbb{Z}[sqrt{-5}]), then either (a) or (b) must be divisible by 3, since 3 is prime in (mathbb{Z}). The same reasoning applies to 7.

For (4 - sqrt{-5}) and (4 sqrt{-5}):

To show that (4 - sqrt{-5}) is prime, assume (4 - sqrt{-5} mid ab) for some (a, b in mathbb{Z}[sqrt{-5}]). Then:

[ N(4 - sqrt{-5}) 21 ]

If (4 - sqrt{-5}) divides (ab), then (N(4 - sqrt{-5}) 21) must divide (N(ab) N(a)N(b)). Therefore, 21 must divide (N(a)N(b)).

The possible norms of elements in (mathbb{Z}[sqrt{-5}]) can be calculated as (m^2 - 5n^2) for (a m nsqrt{-5}). The only pairs (N(a) N(b)) that multiply to a multiple of 21 must also be such that at least one of them is divisible by 21. Since the only norms that are multiples of 21 in (mathbb{Z}) are 21 and higher, if (N(a) 21), then (N(b)) must be at least 21 to satisfy the product condition, which implies that either (a) or (b) must be divisible by (4 - sqrt{-5}).

A similar argument applies to (4 sqrt{-5}), indicating that both elements are prime in (mathbb{Z}[sqrt{-5}]).

Conclusion

Therefore, 3, 7, (4 - sqrt{-5}), and (4 sqrt{-5}) are all prime elements in (mathbb{Z}[sqrt{-5}]).