Algebra

suedenation · #1 $Old$ April 10th, 2006, 09:52 PM

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.....

CaptainBlack · #2 $Old$ April 21st, 2006, 05:57 AM

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

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

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

RonL

CaptainBlack · #3 $Old$ April 21st, 2006, 06:02 AM

Quote:

Originally Posted by suedenation

or not an angle of 6 degree is constructible

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

$60=4.3.5=2^2.3.5$

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

RonL