# Algebra

• Apr 10th 2006, 08:52 PM
suedenation
Algebra
1.Use Gauss Theorem to prove that an angle of 20 degree is not contructible.

2.Use Gauss Theorem to decide whether or not an angle of 6 degree is constructible.

Thanks very much guys..... :)
• Apr 21st 2006, 04:57 AM
CaptainBlack
Quote:

Originally Posted by suedenation
[COLOR=Green]1.Use Gauss Theorem to prove that an angle of 20 degree is not contructible.

Gauss theorem states that an $\displaystyle n-gon$ is constructible $\displaystyle \mbox{iff}$ $\displaystyle n=2^k.p_1,\dots, p_r$,
where $\displaystyle k \in \mathbb{Z_+}$ and $\displaystyle p_1,\dots, p_r$ are distinct Fermat primes (note $\displaystyle r$ can be $\displaystyle 0$).

If an angle of $\displaystyle 20$ degrees were constructible so would a $\displaystyle 360/20=18$ sided
polygon.

The first three Fermat primes are $\displaystyle 3,\ 5,\ 17$, clearly $\displaystyle 5$ and $\displaystyle 17$
do not divide $\displaystyle 18$, so for the $\displaystyle 18-gon$ to be constructible
$\displaystyle 18$ would have to be a power of $\displaystyle 2$, or $\displaystyle 6$ (as it is $\displaystyle 18/3$) would
have to be a power of $\displaystyle 2$. They are not so the $\displaystyle 18-gon$ is not
constructible and so an angle of 20 degrees is not constructible.

RonL
• Apr 21st 2006, 05:02 AM
CaptainBlack
Quote:

Originally Posted by suedenation
or not an angle of 6 degree is constructible

The $\displaystyle 6$ degree angle is constructible $\displaystyle \mbox{iff}$
the $\displaystyle 360/6=60$ sided polygon is constructible.

$\displaystyle 60=4.3.5=2^2.3.5$

so the $\displaystyle 60-gon$ is constructible and so the $\displaystyle 6$ degree
angle is constructible.

RonL