I have been trying to follow a proof of the Abel Ruffini theorem as described in "Mathematical Omnibus", which starts with a quintic in the Bring Gerrard form. However, the only things that I could find on the reduction of the General quintic to the Bring Gerrard form without using resultants, something that I want to avoid as they look very complicated. I was wondering if there was a substitution that could be made along the lines of how Cardano solved the depressed cubic that could transform the quintic into the Bring Gerrard form. Any help would be greatly appreciated. Thanks!
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