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Galois-theoretic derivation of the quartic formula

resolvent cubic
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12D10 no label found


I've been trying to write a program that does this automatically for my calculator, but have encountered a problem. Which root I choose from the resolvent cubic to be root 1 (t1) affects the result, and it's a different root that's needed for some equations than in others.

I solve the resolvent using the equations from "galois theoretic derivation of the cubic equation", so my t1 is r1 from that page, t2 = r2 etc.

Is there any way to know which root yields the correct result?

I wonder if there is a more simple but concise expression for the roots. Even for cubic equations, the roots are more simpler than this.

I'm getting the same problem--any help would be most appreciated.

I have noticed this: the root of the resolvent cubic that I choose makes a difference in which roots are defined as r1,r2,r3,r4. I solve for values of (r1+r2), (r3+r4), (r1*r2), and (r3*r4) that all look reasonable. It's just that sometimes I have a mislabeling. r1 + r2 = (r1+r2), as expected, but r1 * r2 = (r3*r4), and not (r1*r2). When you attempt to solve the resulting quadratics, chaos ensues--you're not solving ones that are meaningful in any way.

These Galois-theoretic derivations are really neat, but there's still some ambiguity in them that's keeping me from programming them algorithmically.

Thank you. :)

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