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???steps to factorize: a^2-ab+b^2-bc+c^2-ca watch

1. I don't know the procedure to factorize:

into

is complex cube root of unity:

================================ =============
Can all the quadratic forms be factorized with complex roots?

2. You need to pull a rabbit out of a hat.

Show that .

Spoiler:
Show

Either from the norm in ,

or by simple algebra starting with and .

Now, you have , so that

Spoiler:
Show

and

and you easily observe that

and

since .

(the intuition being [at least for integers] that is a UFD)

The symmetry of the quadratic form also leads to , etc.
(you can get the same by multiplying through by , but that's not the point.)

(Original post by BookInquiry)
Can all the quadratic forms be factorized with complex roots?
I can answer your question trivially, since these can be viewed as quadratic equations.
However, this does not mean that you would get a nice symmetric expression as above.
3. Truly, I don't understand the method you introduced. I think before I am able to understand your words, there are many topics I should go to study. e.g. The meaning of the term "norm".
Jack, could you suggest some books best discussing about this topic?
4. (Original post by BookInquiry)
I think before I am able to understand your words, there are many topics I should go to study. e.g. The meaning of the term "norm".
I replied to your private message, but I wouldn't jump immediately on studying those topics.
In what context did you find this quadratic form? What is your background knowledge?

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