Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 2 / a / iv

Table of contents
- Solution iv

Solution

For

x \in L

, choose a monic equation over

K

,

x^{n} + c_{1} x^{n - 1} + \dots + c_{n} = 0.

Choose nonzero

a \in R

with

a c_{i} \in R

for every

i

. Multiplying by

a^{n}

shows that

b = a x

satisfies

b^{n} + (a c_{1}) b^{n - 1} + (a^{2} c_{2}) b^{n - 2} + \dots + a^{n} c_{n} = 0,

whose coefficients lie in

R

. Thus

b \in \overline{R}

and

x = b / a, b \in \overline{R}, a \in R .

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version