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http://www.mccme.ru/mmks 21.12.2014,
13.30-13.40, =-. . . . 13.40-14.15, . .. . . ( .. ) 14.20-14.35, . . . ( .. ) 14.40-15.15, . . Solvability of cubic and quartic equations using one radical. ( .. ) 15.15-15.45, 404. (, , ) 15.20-15.45. ( : .. ) 15.20-15.45 ( ) ( ) 404. 13.20. . , . . . 15.45-16.20, . . . ( .. ) 16.25-16.50, . . , k . ( .. ) 16.55-17.30, . .. . . ( .. ) 17.30-17.45, . 404. (, , ) 17.45-17.55, . 18.00-18.30, . 18.00-19.00, . 404. .. 18.00-19.00, . 304. ..


-2014
. http://www.mccme.ru/mmks/notes.htm, notesm.htm
. . , . . . . , 1 4, , . . , -. . , . , . `' ( , ), `'. . - . Solvability of cubic and quartic equations using one radical. Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x4 + px2 + q x + s with rational coefficients q = 0, p and s has a root in a one step radical extension of Q if and only if the cubic resolution has rational root t such that A := 16(t2 - s)2 - (t2 - s)(2t + p)2 is a square of a rational number. These theorems are proved by Chu and Kang in 2000-2002. In this note we present shorter proofs of the `only if ' parts. Our proofs are based on expressions of the discriminate, of the roots of the cubic resolution and of A in terms of the roots.


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