Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am reading Rational Points on Elliptic Curves by Silverman and Tate, and on page 48 authors make the following remark: (bold emphasis mine)

If $f(x)$ is any polynomial with leading coefficient $1$ in the ring $\mathbb{Z}[x]$ of polynomials with integer coefficients, then the discriminant of $f(x)$ will always be in the ideal of $\mathbb{Z}[x]$ generated by $f(x)$ and $f'(x)$. This follows from the general theory of discriminants, but for our particular polynomial $f(x) = x^3 + ax^2 + bx + c$, the quickest proof is just to write out an explicit formula:

$$D = \left\{(18b - 6a^2)x - (4a^3 - 15ab + 27c)\right\} f(x) +\left\{(2a^2- 6b)x^2+(2a^3 - 7ab+9c)x+(a^2b+3ac- 4b^2)\right\}f'(x)$$ We leave it to you to multiply this out and verify it is correct.

Could someone show how this statement follows from the general theory of discriminants? I would be delighted to learn :)

share|improve this question

1 Answer 1

up vote 5 down vote accepted

This follows from two basic facts that can be found in most books:

  1. The discriminant (up to sign!) of $f$ is the resultant of $f$ and $f'$.
  2. If $P$ and $Q$ are any two polynomials, there always exists two polynomials $A$ and $B$ such that $AP+BQ=\text{res}(P,Q)$.
share|improve this answer
    
@YACP Of course, I edited accordingly. Thank you! –  Alex Youcis Aug 10 '13 at 17:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.