MathDB

Let

Z=\{ z_1,\ldots, z_{n-1}\}

n\ge 2

, be a set of different complex numbers such that

Z

contains the conjugate of any its element. a) Show that there exists a constant

C

, depending on

Z

, such that for any

\varepsilon\in (0,1)

there exists an algebraic integer

x_0

of degree

n

, whose algebraic conjugates

x_1, x_2, \ldots, x_{n-1}

satisfy

|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon

and

|x_0|\le \frac{C}{\varepsilon}

. b) Show that there exists a set

Z=\{ z_1, \ldots, z_{n-1}\}

and a positive number

c_n

such that for any algebraic integer

x_0

of degree

n

, whose algebraic conjugates satisfy

|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon

, it also holds that

|x_0|>\frac{c_n}{\varepsilon}

.
(translated by L. Erdős)

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