MHB Proof of Infinitely Many Units of Z[sqrt n] for Eacn Non-Square Natural Number

[Math Amateur]

In John Stillwell's book: Elements of Number Theory, Exercise 6.1.3 reads as follows:

-------------------------------------------------------------------------------------------------

Show that $$\mathbb{Z} [ \sqrt{n}$$ has infinitely many units for any non-square natural number n

-------------------------------------------------------------------------------------------------

Can someone please help me get started on this exercise?

Peter

 

[Euge]

In John Stillwell's book: Elements of Number Theory, Exercise 6.1.3 reads as follows:

-------------------------------------------------------------------------------------------------

Show that $$\mathbb{Z} [ \sqrt{n}$$ has infinitely many units for any non-square natural number n

-------------------------------------------------------------------------------------------------

Can someone please help me get started on this exercise?

Peter

Hint: Apply Dirchlet's unit theorem to $\Bbb Z[\sqrt{n}]$.

 

[mathbalarka]

Hint: Apply Dirchlet's unit theorem to $\Bbb Z[\sqrt{n}]$.

Another relatively diophantine approach would be to look at $x^2 - ny^2 = \pm 1$. Appropriate diophantine approximations on $\sqrt{n}$ can produce at least one solution to $x^2 - ny^2 = 1$, thus producing infinitely many of them by fundamental results on Pell equation.

 

[Math Amateur]

Hint: Apply Dirchlet's unit theorem to $\Bbb Z[\sqrt{n}]$.

Thanks Euge ...

Will have to read up on Dirichlet's Unit Theorem ...

Peter

- - - Updated - - -

Another relatively diophantine approach would be to look at $x^2 - ny^2 = \pm 1$. Appropriate diophantine approximations on $\sqrt{n}$ can produce at least one solution to $x^2 - ny^2 = 1$, thus producing infinitely many of them by fundamental results on Pell equation.

Thanks Mathbalarka ...

Will have to read up on The Pell equation to be able to follow your advice ...

Peter