By Bandini A.
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Extra info for 3-Selmer groups for curves y^2 = x^3 + a
Nd carried that see all are restrictions no on However, in the case two complex numbers for the determine calculations we since general, other, pairs of cube roots conjugate to each other; only in also are furthermore, $1, in in the 12 u this be satisﬁed. ], = : for j to calculate = solutions For 1, 2, 3. That with are the real : is, in employing Cardano’s formula, it is even in the case that complex numbers and problem Cardano’s Ars = necessary all three distinct. 19\F 1 643 2 1 =3. , 5 2 1 3 , 5 of The 2.
However, if viewed at correctly, this result is not as surprising as it might appear ﬁrst glance. In fact, we encountered something comparable in the case of cubic equations: Just as Cardano’s formula contains roots square in addition to cube roots, a general formula for biquadratic equations must be similarly constituted. 2 \xC0\x9C (131 = Ferrari’s formula Since presented as biquadratic equations wwmmwmxmave in which third biquadratic describe must we power, the of the equation a here Variable as of method can does be used not only to appear converting the for the general form w4+aa:3+b:t2+cm+d=0 into an in the equation reduced form y"+py2+qy+r=0In analogy to replacing the the of the case variable av via cubic the equation, this be can done by substitution a ‘T:ywZ> with the result that the two terms in that ye‘ arise cancel each other: a:4+cLa:3+bac2+c$+d=y4+py2-l-qy+7*.
Solution a surprising at always starts as method polynomials in at ﬁrst might seem values intermediate express as it with the coefficients general equation with in the solutions, it is actually the of be expressed in terms that all the intermediate values can obvious the solutions operations and nested roots. using the usual arithmetic However, What is not a priori obvious is that polynomials alone suﬁice, That for degrees two, three, and four. the case which was is, in the of the intermediate values, no expression of the form, representation equation, that is, in reference elementary symmetric polynomials of the to the Say, V‘/$1+ may \xF0\x88c appear.