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Euler's derivation of the quartic formula
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(Theorem)
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Let us consider the quartic equation
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(1) |
where $p,\,q,\,r$ are arbitrary known complex numbers. We substitute in the equation
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(2) |
We get firstly
$y^2 = (u^2+v^2+w^2)+2(vw+wu+uv),$
$y^4 = (u^2+v^2+w^2)^2+4(u^2+v^2+w^2)(vw+wu+uv)+4(v^2w^2+w^2u^2+u^2v^2)+8uvw(u+v+w).$
Thus (1) attains the form $$4(v^2w^2+w^2u^2+u^2v^2)+(u^2+v^2+w^2)^2+p(u^2+v^2+w^2)+r \qquad\;$$ $$+(vw+wu+uv)[4(u^2+v^2+w^2)+2p]+(u+v+w)[8uvw+q] = 0.$$ When $u,\,v,\,w$ are determined so that
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(3) |
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(4) |
the expressions in the brackets vanish and our equation shrinks to the form
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(5) |
Squaring (4) gives
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(6) |
The left hand sides of (3), (5) and (6) are the elementary symmetric polynomials of $u^2$ , $v^2$ , $w^2$ , whence these three squares are the roots $z_1$ , $z_2$ , $z_3$ of the so-called cubic resolvent equation
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(7) |
Therefore we may write $$u = \pm\sqrt{z_1}, \quad v = \pm\sqrt{z_2}, \quad w = \pm\sqrt{z_3}.$$ All 8 sign combinations of those square roots satisfy the equations (3), (5), (6). In order to satisfy also (4) the signs must be chosen suitably. If $u_0,\,v_0,\,w_0$ is some suitable combination of the values of the square roots, then all possible combinations are $$u_0,\,v_0,\,w_0;\quad u_0,\,-v_0,\,-w_0;\quad -u_0,\,v_0,\,-w_0;\quad -u_0,\,-v_0,\,w_0.$$
Accordingly, we have the
Theorem (Euler 1739). The roots of the equation (1) are
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(8) |
where $u_0,\,v_0,\,w_0$ are square roots of the roots of the cubic resolvent (7). The signs of the square roots must be chosen such that $$u_0v_0w_0 = -\frac{q}{8}.\\$$
The equations (8) imply an important formula
which yields the
Corollary. A quartic equation has a multiple root always and only when its cubic resolvent has such one.
- 1
- ERNST LINDELÖF: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
- 2
- K. V¨AISÄLÄ: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
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"Euler's derivation of the quartic formula" is owned by pahio.
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Cross-references: multiple root, formula, imply, Euler, theorem, combination, order, square roots, combinations, cubic resolvent, roots, squares, elementary symmetric polynomials, left hand sides, vanish, expressions, equation, complex numbers, quartic equation
This is version 7 of Euler's derivation of the quartic formula, born on 2008-02-27, modified 2008-03-03.
Object id is 10344, canonical name is EulersDerivationOfTheQuarticFormula.
Accessed 2016 times total.
Classification:
| AMS MSC: | 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros ) |
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Pending Errata and Addenda
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