arqady wrote:

Let $a$ , $b$ and $c$ be roots of the equation $x^3-16x^2-57x+1=0$ . Prove that:
$\sqrt[5]a+\sqrt[5]b+\sqrt[5]c=1$

arqady, I really want to know your solution, because my is very ugly and done by software.
$a\rightarrow a^5$ , $b\rightarrow b^5$ , $c\rightarrow c^5$ .
Let $3u=a+b+c$ , $3v^2=ab+bc+ca$ ( $v^2$ can be negative) and $w^3=abc$ .
$\begin{cases}243u^5-405u^3v^2+135uv^4+(45u^2-15v^2)w^3=16\\243v^{10}-405uv^6w^3+135u^2v^2w^6+45v^4w^6-15uw^9=-57\\w^{15}=-1\end{cases}$
Since $w^3=-1$ , then
$135uv^4-(405u^3-15)v^2+243u^5-45u^2-16=0$
$v^2=\frac{135u^3\pm\sqrt{5(729u^6+270u^3+192u+5)}}{90u}$
For clearance $x=v^2$ .
Case $x=\frac{135u^3-\sqrt{5(729u^6+270u^3+192u+5)}}{90u}$ , by maxima

expand(subst ([x=(135*u^3-5-sqrt(5*(729*u^6+270*u^3+192*u+5)))/(90*u)], 81*x^5+135*u*x^3+15*x^2+45*u^2*x+5*u+19));
ratsimp(%);

Gives $14348907\sqrt{5^3}u^{15}+7440174\sqrt{5^3}u^{12}+2834352\sqrt{5^3}u^{10}+39366\sqrt{5^7}u^9+ 419904\sqrt{5^3}u^7+4374\sqrt{5^5}u^6+200718\sqrt{5^3}u^5-2304\sqrt{5^3}u^2-48\sqrt{5^5}u-\sqrt{5^5}=(5845851u^{12}+1869885u^9+454896u^7+98415u^6+6480u^4- 675u^3+2304u^2+720u+25)\sqrt{729u^6+270u^3+192u+5}$
squarring
$(14348907\sqrt{5^3}u^{15}+7440174\sqrt{5^3}u^{12}+2834352\sqrt{5^3}u^{10}+39366\sqrt{5^7}u^9+ 419904\sqrt{5^3}u^7+4374\sqrt{5^5}u^6+200718\sqrt{5^3}u^5-2304\sqrt{5^3}u^2-48\sqrt{5^5}u-\sqrt{5^5})^2=(5845851u^{12}+1869885u^9+454896u^7+98415u^6+6480u^4- 675u^3+2304u^2+720u+25)^2(729u^6+270u^3+192u+5)$
$LHS-RHS$
$823564528378596u^{30}+1525119496997400u^{27}-271132355021760u^{25}+988503377683500u^{24}-150629086123200u^{22}+ 282429536481000u^{21}+532571449408740u^{20}-9298091736000u^{19}+35836395232500u^{18}+247716660666600u^{17}+4476858984000u^{16}+ 97373978727120u^{15}+37751974317000u^{14}-1211685480000u^{13}+16083902432700u^{12}-671209983000u^{11}+4516257657780u^{10}-486189783000 u^9-75582720000u^8-126279129600u^7-14328130500u^6-1347873372u^5=972u^5(3u-1)(531441u^{12}+531441u^{11}+354294u^{10}+688905u^9+885735u^8+509571u^7+ 94041u^6+96714u^5+106515u^4+36180u^3+10044u^2+1029u+71)(531441u^{12}- 354294u^{11}+59049u^{10}+492075u^9-623295u^8+236196u^7+305451u^6-360126u^5+ 336150u^4-178335u^3+109719u^2-16851u+19531)=0$ . Obviously $u=0$ is not solution.

allroots(531441*u^12+531441*u^11+354294*u^10+688905*u^9+885735*u^8+509571*u^7+
94041*u^6+96714*u^5+106515*u^4+36180*u^3+10044*u^2+1029*u+71);

Gives

[u=0.08873583372698096*%i-0.06285164678937094,u=-0.08873583372698096*%i-0.06285164678937094,
u=0.249818724727736*%i-0.1038150198772957,
u=-0.249818724727736*%i-.1038150198772957,u=0.4141258170774749*%i+0.40187021328863,
u=0.40187021328863-0.4141258170774749*%i,u=0.04493113182149037*%i-0.7343941915363165,
u=-0.04493113182149037*%i-0.7343941915363165,u=0.6072801786518509*%i-0.5685368799552993,
u=-0.6072801786518509*%i-.5685368799552993,u=0.9566536050241474*%i+0.5677275248696524,
u=0.5677275248696524-0.9566536050241474*%i]

allroots(531441*u^12-354294*u^11+59049*u^10+492075*u^9-623295*u^8+236196*u^7+305451*u^6-360126*u^5+
336150*u^4-178335*u^3+109719*u^2-16851*u+19531);

Gives

[u = 0.4301615523642846 %i - 0.1842082400516285, 
u = (- 0.4301615523642846 %i) - 0.1842082400516285, 
u = 0.5973695492100051 %i - 0.001309564037543536, 
u = (- 0.5973695492100051 %i) - 0.001309564037543536, 
u = 0.5364026821017057 %i + 0.5180417823368586, 
u = 0.5180417823368586 - 0.5364026821017057 %i, 
u = 0.3315150891954592 %i + 0.81690633083384, 
u = 0.81690633083384 - 0.3315150891954592 %i, 
u = 0.9665642344659907 %i + 0.2055111206314012, 
u = 0.2055111206314012 - 0.9665642344659907 %i, 
u = 0.2658544600145461 %i - 1.021608096379594, 
u = (- 0.2658544600145461 %i) - 1.021608096379594]

Case $x=\frac{135u^3-5+\sqrt{5(729u^6+270u^3+192u+5)}}{90u}$ , since $u^2\geq v^2$ , then $u\leq -1.15804$
by maxima

ratsimp(ratsimp(expand(subst ([x=(135*u^3-5+sqrt(5*(729*u^6+270*u^3+192*u+5)))/(90*u)], 81*x^5+135*u*x^3+15*x^2+45*u^2*x+5*u+19))));

$14348907\sqrt{5^3}u^{15}+7440174\sqrt{5^3}u^12+2834352\sqrt{5^3}u^10+39366\sqrt{5^7}u^9 +419904\sqrt{5^3}u^7+4374\sqrt{5^5}u^6+200718\sqrt{5^3}u^5-2304\sqrt{5^3}u^2-48\sqrt{5^5}u-\sqrt{5^5}=-\sqrt{729u^6+270u^3+192u+5} (5845851u^{12}+1869885u^9+454896u^7+98415u^6+6480u^4-675u^3+2304u^2+720u+25)$
Squrring
$(14348907\sqrt{5^3}u^{15}+7440174\sqrt{5^3}u^12+2834352\sqrt{5^3}u^10+39366\sqrt{5^7}u^9 +419904\sqrt{5^3}u^7+4374\sqrt{5^5}u^6+200718\sqrt{5^3}u^5-2304\sqrt{5^3}u^2-48\sqrt{5^5}u-\sqrt{5^5})^2=(5845851u^{12}+1869885u^9+454896u^7+98415u^6+6480u^4-675u^3+2304u^2+720u+25)^2(729u^6+270u^3+192u+5)$
$LHS-RHS$
$-24912826983452529u^{30}-25164471700457100u^{27}-10438595668337760u^{25}-2541865828329000u^{24}-1656919947355200u^{22}+ 203976887458500u^{21}-187449529397760u^{20}-9298091736000u^{19}+35836395232500u^{18}+255981631098600u^{17}+5337793404000u^{16}+ 97391914860870u^{15}+37751974317000u^{14}-1211685480000u^{13}+26705675099887200u^{12}-671209983000u^{11}+10171979570973780u^{10}+ 3529883016229500u^9-75582720000u^8+1506164582102400u^7+78438320892000u^6+720019630933128u^5-8264970432000u^2-860934420000 u+25736373575697375=0$
Again typing in maxima

solve(-24912826983452529*u^30-25164471700457100*u^27-10438595668337760*u^25-2541865828329000*u^24-1656919947355200*u^22+
203976887458500*u^21-187449529397760*u^20-9298091736000*u^19+35836395232500*u^18+255981631098600*u^17+5337793404000*u^16+
97391914860870*u^15+37751974317000*u^14-1211685480000*u^13+26705675099887200*u^12-671209983000*u^11+10171979570973780*u^10+
3529883016229500*u^9-75582720000*u^8+1506164582102400*u^7+78438320892000*u^6+720019630933128*u^5-8264970432000*u^2-860934420000*
u+25736373575697375,u);

Gives

[[0=102521921742603*u^30+103557496709700*u^27+42957183820320*u^25+10460353203000*u^24+6818600606400*u^22-839411059500*
u^21+771397240320*u^20+38263752000*u^19-147474877500*u^18-1053422350200*u^17-21966228000*u^16-400789773090*u^15-155357919000*u^14+
4986360000*u^13-109899897530400*u^12+2762181000*u^11-41859998234460*u^10-14526267556500*u^9+311040000*u^8-6198208156800*u^7-
322791444000*u^6-2963043748696*u^5+34012224000*u^2+3542940000*u-105911002369125]=0]

typing

allroots(102521921742603*u^30+103557496709700*u^27+42957183820320*u^25+10460353203000*u^24+6818600606400*u^22-839411059500*
u^21+771397240320*u^20+38263752000*u^19-147474877500*u^18-1053422350200*u^17-21966228000*u^16-400789773090*u^15-155357919000*u^14+
4986360000*u^13-109899897530400*u^12+2762181000*u^11-41859998234460*u^10-14526267556500*u^9+311040000*u^8-6198208156800*u^7-
322791444000*u^6-2963043748696*u^5+34012224000*u^2+3542940000*u-105911002369125,u);

gives

u = 0.6701030312889549 - 0.7418603072941804 %i, 
u = 0.6936763996501353 %i - 0.6394643083013833, 
u = (- 0.6936763996501353 %i) - 0.6394643083013833, 
u = 0.8693833939058644 %i - 0.5003277841706911, 
u = (- 0.8693833939058644 %i) - 0.5003277841706911, u = 1.003024944024268, 
u = 0.9515836439685664 %i + 0.308918723378388, 
u = 0.308918723378388 - 0.9515836439685664 %i, 
u = 0.5896639595395451 %i - 0.8091698531500543, 
u = (- 0.5896639595395451 %i) - 0.8091698531500543, 
u = 0.9340675629015904 %i - 0.2681691050314659, 
u = (- 0.9340675629015904 %i) - 0.2681691050314659, 
u = 0.9930125376811316 %i + 0.1466433765516465, 
u = 0.1466433765516465 - 0.9930125376811316 %i, 
u = 0.9950264234389671 %i - 0.1076485022948794, 
u = (- 0.9950264234389671 %i) - 0.1076485022948794, 
u = 0.3220753007654913 %i - 0.9341384895486041, 
u = (- 0.3220753007654913 %i) - 0.9341384895486041, 
u = 0.2386333647302399 %i + 0.9080121402521805, 
u = 0.9080121402521805 - 0.2386333647302399 %i, u = - 1.132184434639223, 
u = 0.9072105915610396 %i + 0.5644742490679983, 
u = 0.5644742490679983 - 0.9072105915610396 %i, 
u = 0.6763316638574209 %i + 0.7886132313994743, 
u = 0.7886132313994743 - 0.6763316638574209 %i, 
u = 0.2073934966713666 %i - 0.9781321310923142, 
u = (- 0.2073934966713666 %i) - 0.9781321310923142, 
u = 0.406721388877138 %i + 0.9148651669582274, 
u = 0.9148651669582274 - 0.406721388877138 %i]

So only $u=\frac{1}{3}$ is real solution