Law of inertia for the factorization of cubic polynomials - the imaginary case
| Authors | |
|---|---|
| Year of publication | 2017 |
| Type | Article in Periodical |
| Magazine / Source | Utilitas Mathematica |
| MU Faculty or unit | |
| Citation | |
| Web | http://91.203.202.198/view/j/ms.2017.67.issue-1/ms-2016-0248/ms-2016-0248.xml |
| Field | General mathematics |
| Keywords | cubic polynomial; type of factorization; discriminant |
| Description | Let D be a square-free positive integer not divisible by 3 such that the class number h(-3D) of Q((-3D)^(1/2)) is also not divisible by 3. We prove that all cubic polynomials f (x) = x^3 + ax^2 + bx + c in Z[x] with a discriminant D have the same type of factorization over any Galois field F_p, where p is a prime bigger than 3. Moreover, we show that any polynomial f(x) with such a discriminant D has a rational integer root. A complete discussion of the case D = 0 is also included. |
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