Proving Prime Norms of H: (a,b) in Z

[Hud123]

H is a number system where (a,b) belongs in H where a and b is an element of Z(integer)

Addition and multiplication are defined as follows:

(a, b) + (c, d) := (a + c , b + d)

(a, b) x (c, d) := (ac-5bd , ad+bc)

For any number (a,b) in H, we can define its norm by

||(a, b)|| := a^2+5*(b^2)

Prove that (a, b) is prime in H if ||(a, b)|| is prime in N(Natural numbers)

I'm having a bit of trouble where to start for this proof, can anyone help?

Also how could I show say (5,0) is not prime in H while it is in N?

Thank you in advance

 

[Opalg]

H is a number system where (a,b) belongs in H where a and b is an element of Z(integer)

Addition and multiplication are defined as follows:

(a, b) + (c, d) := (a + c , b + d)

(a, b) x (c, d) := (ac-5bd , ad+bc)

For any number (a,b) in H, we can define its norm by

||(a, b)|| := a^2+5*(b^2)

Prove that (a, b) is prime in H if ||(a, b)|| is prime in N(Natural numbers)

I'm having a bit of trouble where to start for this proof, can anyone help?

Also how could I show say (5,0) is not prime in H while it is in N?

Thank you in advance

Hi Hud and welcome to MHB!

The key property of a norm in a number system is that it should satisfy the condition $\|(a,b)\times (c,d)\| = \|(a,b)\|\,\|(c,d)\|.$ So your first job should be to check that this multiplicative condition holds here. Namely, you need to check that $(ac-5bd)^2 + 5(ad+bc)^2 = (a^2 + 5b^2)(c^2 + 5d^2)$. Once you have done that, you need to figure out why that property implies that a prime in $H$ is necessarily a prime in $\mathbb{N}$.

That property also tells you that if $(5,0)$ is equal to a product in $H$, say $(5,0) = (a,b)\times (c,d)$, then $\|(a,b)\|\,\|(c,d)\| = \|(5,0)\| = 25$. Since $25 = 5\times5$, it looks as though you should be trying to find elements $(a,b)$ and $(c,d)$ in $H$ with norm $5$.

 

[Hud123]

Thank you for the reply, I have proved that the multiplicative holds but I am not too certain what property I should be looking at?