Solving Quadratic Field Problems with Quick Tips - PhysicsForums

[Brimley]

Hello PhysicsForums!

I had posted a question earlier today in another thread and I had a follow up question to it (the question in nature isn't extremely related, but the link can be found https://www.physicsforums.com/showthread.php?p=2964009#post2964009").

In the previous example, it was listed that $\lambda = (3+\sqrt{-3})/2 \in \mathbb{Q}[\sqrt{3}]$.

The text states the following:

"$x,y,$ and $z$ are quadratic integers in $\mathbb{Q}[\sqrt{-3}]$, where $x^3 + y^3 = z^3$." From here it can be shown that $\lambda$ can divide one of $x,y,$ or $z$.

Can anyone help explain this? I don't know if reducing the equation modula $\lambda^3$ would help, but its my first guess.

Thanks -- Brim

 

[hochs]

Hello PhysicsForums!

I had posted a question earlier today in another thread and I had a follow up question to it (the question in nature isn't extremely related, but the link can be found https://www.physicsforums.com/showthread.php?p=2964009#post2964009").

In the previous example, it was listed that $\lambda = (3+\sqrt{-3})/2 \in \mathbb{Q}[\sqrt{3}]$.

The text states the following:

"$x,y,$ and $z$ are quadratic integers in $\mathbb{Q}[\sqrt{-3}]$, where $x^3 + y^3 = z^3$." From here it can be shown that $\lambda$ can divide one of $x,y,$ or $z$.

Can anyone help explain this? I don't know if reducing the equation modula $\lambda^3$ would help, but its my first guess.

Thanks -- Brim

What book are you reading anyway? And you're trying to learn algebraic number theory without solid background in abstract algebra, particularly galois theory. Of course you run into troubles.

If you're not doing these for homework and have no time-constraint, then I suggest you get some solid background in abstract algebra first, then read good intro books like cassels & frohlich's algebraic number theory, neukirch's algebraic number theory, lang's, etc.

hint: you can factor x^3 + y^3 = (x + y)(x + yw)(x + yw^2), where w is the primitive 3rd root of unity, and gcd(x+y, x+yw^i) = gcd(x+y, 1 - w) for any i != 0 (mod 3) (direct computation). Also, 3 totally ramifies in the ring Z[w] as (3) = (1 - w)^2.

The above remarks are all one needs. I will not say more than the above hint, but others can feel free to expand / give more details.

 

[Brimley]

What book are you reading anyway? And you're trying to learn algebraic number theory without solid background in abstract algebra, particularly galois theory. Of course you run into troubles.

If you're not doing these for homework and have no time-constraint, then I suggest you get some solid background in abstract algebra first, then read good intro books like cassels & frohlich's algebraic number theory, neukirch's algebraic number theory, lang's, etc.

hint: you can factor x^3 + y^3 = (x + y)(x + yw)(x + yw^2), where w is the primitive 3rd root of unity, and gcd(x+y, x+yw^i) = gcd(x+y, 1 - w) for any i != 0 (mod 3) (direct computation). Also, 3 totally ramifies in the ring Z[w] as (3) = (1 - w)^2.

The above remarks are all one needs. I will not say more than the above hint, but others can feel free to expand / give more details.

Hochs, I would like to finish this same example so I can have it to reference when I do read up on things, however it is kind of pointless for me to have come this far with this same problem to quit near the end, spend months reading up, then having to come back and revisit all of this and resurrect a dead thread. For this reason, I would really like to finish this example.

Back to the problem: I know that $\lambda = 1 - w = (3+\sqrt{-3})/2$. Looking at your factoring and use of the GCD, the lost variable here is the $z^3$. Where does this play in? Which one does $\lambda$ then divide into?