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Iran MO (3rd Round)
2012 Iran MO (3rd Round)
3
p-th root of rational numbers
p-th root of rational numbers
Source: Iran 3rd round 2012-Algebra exam-P3
September 20, 2012
algebra
polynomial
algebra proposed
Problem Statement
Suppose
p
p
p
is a prime number and
a
,
b
,
c
∈
Q
+
a,b,c \in \mathbb Q^+
a
,
b
,
c
∈
Q
+
are rational numbers;a) Prove that
Q
(
a
p
+
b
p
)
=
Q
(
a
p
,
b
p
)
\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})
Q
(
p
a
+
p
b
)
=
Q
(
p
a
,
p
b
)
.b) If
b
p
∈
Q
(
a
p
)
\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})
p
b
∈
Q
(
p
a
)
, prove that for a nonnegative integer
k
k
k
we have
b
a
k
p
∈
Q
\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q
p
a
k
b
∈
Q
.c) If
a
p
+
b
p
+
c
p
∈
Q
\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q
p
a
+
p
b
+
p
c
∈
Q
, then prove that numbers
a
p
,
b
p
\sqrt[p]{a},\sqrt[p]{b}
p
a
,
p
b
and
c
p
\sqrt[p]{c}
p
c
are rational.
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