1
$\begingroup$

consider $\mathbb{Q}\subset K$ a finite algebraic extension. Take $x\in K$ integral, why $\mid Norm_{K/\mathbb{Q}}(x)\mid \geq 1$?

Another question is: is it true that $\bar{\mathbb{Q}}_p \cong \mathbb{C}$? if it is so why?

Thank you.

  • 0
    yes, it is algebraic closure2011-04-18

1 Answers 1

1

The norm of an integer is a rational integer.

$\mathbb Q_p$ cannot be extended to $\mathbb C$ because it has a different metric.

  • 0
    To expand a little on quanta's answer. $x\in K$ integral means $x$ is a root of a monic polynomial with integer coefficients. Its norm is the product of its conjugates, which is (up to sign) the constant term of said monic polynomial.2011-04-18