Is \(\mathbb{Q}(\sqrt{5})\) a Unique Factorization Domain?

[math_grl]

So you see it all over the place, $$\mathbb{Q}(\sqrt{-5})$$ is not a UFD by finding an element such that it has two distinct prime factorizations...but what about showing that $$\mathbb{Q}(\sqrt{5})$$ is a UFD?

I'm only concerned with this particular example, I might have questions later on regarding a general method.

 

[MathematicalPhysicist]

Isn't the proof goes like the proof of the fundamental theorem of arithmetic? the uniqueness part.

 

[math_grl]

I guess it could be. So you are saying that you take some random element $$a + b\sqrt{5} \in \mathbb{Q}(\sqrt{5})$$ and claim there are two distinct prime factorizations and show they actually differ by a unit?

 

[MathematicalPhysicist]

Well if you show it for the integers coeffiecients then it will obviously follow for rational coeffiecnts (I am referring to a,b), by taking common denominator in both sides and then multiply by the lcm of the two denominators of both sides we reducing the problem to integer coeffiecints.

I believe that because Q(\sqrt 5) is spanned by 1 and \sqrt 5
It's enough to show that $$(a+b\sqrt 5) c \sqrt 5=d+e\sqrt 5$$ and $$(a+b\sqrt 5) c =d+e\sqrt 5$$, where all the parameters are integers, is uniquely factorised (this obviously follows from the uniquness in simple integers).
Obviously in the proof we should impose the condition that gcd(a,b,c)=1 otherwise we can divide both sides by gcd(a,b,c).

I am just not sure how this will follow for negative integers, maybe this works upto a sign in the whole integers.

 

[blue_raver22]

I understand the importance of rigorous proof and evidence in mathematical concepts such as unique factorization domains (UFDs). While it is true that \mathbb{Q}(\sqrt{-5}) is not a UFD, it is also important to note that not all quadratic number fields are UFDs. In fact, there are infinitely many quadratic number fields that are not UFDs.

In the case of \mathbb{Q}(\sqrt{5}), we can show that it is indeed a UFD by providing a proof that every nonzero, non-unit element in this field has a unique prime factorization. This can be done by using the fundamental theorem of arithmetic, which states that every positive integer can be uniquely expressed as a product of prime numbers.

In the case of \mathbb{Q}(\sqrt{5}), we can prove that it is a UFD by showing that every nonzero, non-unit element can be written as a product of irreducible elements (elements that cannot be factored further). This can be done by considering the norm of elements in this field, which is defined as the product of the element and its complex conjugate. Using the fact that the norm is multiplicative, we can show that every nonzero, non-unit element in \mathbb{Q}(\sqrt{5}) can be written as a product of irreducible elements.

Furthermore, we can also show that the factorization is unique by assuming that there are two distinct prime factorizations for a given element and then showing that they must be equal. This can be done by using the properties of the norm and the fact that the elements in \mathbb{Q}(\sqrt{5}) have unique factorizations in the Gaussian integers (a UFD).

In conclusion, while it is true that \mathbb{Q}(\sqrt{5}) is a UFD, it is not a general rule that all quadratic number fields are UFDs. Each field must be analyzed individually to determine if it satisfies the properties of a UFD.