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Vietnam National Olympiad
2017 Vietnam National Olympiad
2
Minimal polynomial
Minimal polynomial
Source: 2017 VMO Problem 2
January 11, 2017
algebra
polynomial
Problem Statement
Is there an integer coefficients polynomial
P
(
x
)
P(x)
P
(
x
)
satisfying
{
P
(
1
+
2
3
)
=
1
+
2
3
P
(
1
+
5
)
=
2
+
3
5
\begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases}
{
P
(
1
+
3
2
)
=
1
+
3
2
P
(
1
+
5
)
=
2
+
3
5
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