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    :confused:I don't know the procedure to factorize:

    a^2-ab+b^2-bc+c^2-ca

    into

    (a+\omega b+\omega^2 c)(a+\omega^2 b+\omega c)

    \omega is complex cube root of unity: \omega^3=1

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

    ax^2+2fxy+by^2+2gyz+2px+2qy+d=0
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    You need to pull a rabbit out of a hat.


    Show that b^2 - bc + c^2 = (b + c\omega)(b + c\omega^2).

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    Either from the norm N(a + b\omega) = a^2 - ab + b^2 in \mathbb{Z}[\omega],

    or by simple algebra starting with \alpha + \beta = c and \alpha\beta = c^2.

    Now, you have a^2 - a(b + c) + b^2 - bc + c^2, so that

    Spoiler:
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    \alpha + \beta = b + c and \alpha\beta = (b + c\omega)(b + c\omega^2)

    and you easily observe that

    \alpha = -(b\omega + c\omega^2) and \beta = \overline{\alpha}

    since \omega^2 + \omega + 1 = 0.


    (the intuition being [at least for integers] that \mathbb{Z}[\omega] is a UFD)

    The symmetry of the quadratic form also leads to (c + b\omega + a\omega^2)(c + b\omega^2 + a\omega), etc.
    (you can get the same by multiplying through by \omega^3, 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.
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    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?
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    (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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