Galois theory: Gauss-Wantzel theorem, proof explanation
I am using Ian Stewart Galois theory book
and it says
that for $A = $ primitive $p^2$ root of unity
$A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$
and so $p(p-1)$ is a power of two. why is this so?if $p^k$ -sided regular polygon is constructive, then so must $p^2$ - sided regular polygon, for $k ge 2$. why is this so?
galois-theory proof-explanation minimal-polynomials roots-of-unity geometric-construction
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
add a comment |
I am using Ian Stewart Galois theory book
and it says
that for $A = $ primitive $p^2$ root of unity
$A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$
and so $p(p-1)$ is a power of two. why is this so?if $p^k$ -sided regular polygon is constructive, then so must $p^2$ - sided regular polygon, for $k ge 2$. why is this so?
galois-theory proof-explanation minimal-polynomials roots-of-unity geometric-construction
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
1
I edited your post to make the $LaTeX$ work. Remember, "$" and ${ cdot }$ are your two best friends in $LaTeX$. Cheers!
– Robert Lewis
Dec 2 '18 at 23:22
Did you show that $Phi_p(t) = sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $zeta_p$ ?
– reuns
Dec 3 '18 at 0:12
1. It can’t be that $p(p-1)$ is a power of $2$: Fundamental Theorem of Arithmetic says that in that case, both $p$ and $p-1$ would be powers of $2$. Unless $p-1$=1, of course. For 2, note that not even the $9$-sided regular polygon is constructible.
– Lubin
Dec 3 '18 at 22:05
add a comment |
I am using Ian Stewart Galois theory book
and it says
that for $A = $ primitive $p^2$ root of unity
$A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$
and so $p(p-1)$ is a power of two. why is this so?if $p^k$ -sided regular polygon is constructive, then so must $p^2$ - sided regular polygon, for $k ge 2$. why is this so?
galois-theory proof-explanation minimal-polynomials roots-of-unity geometric-construction
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
I am using Ian Stewart Galois theory book
and it says
that for $A = $ primitive $p^2$ root of unity
$A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$
and so $p(p-1)$ is a power of two. why is this so?if $p^k$ -sided regular polygon is constructive, then so must $p^2$ - sided regular polygon, for $k ge 2$. why is this so?
galois-theory proof-explanation minimal-polynomials roots-of-unity geometric-construction
galois-theory proof-explanation minimal-polynomials roots-of-unity geometric-construction
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
edited Dec 2 '18 at 23:21
Robert Lewis
43.7k22963
43.7k22963
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
asked Dec 2 '18 at 23:11
dkfmaekqwy
6
6
1
I edited your post to make the $LaTeX$ work. Remember, "$" and ${ cdot }$ are your two best friends in $LaTeX$. Cheers!
– Robert Lewis
Dec 2 '18 at 23:22
Did you show that $Phi_p(t) = sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $zeta_p$ ?
– reuns
Dec 3 '18 at 0:12
1. It can’t be that $p(p-1)$ is a power of $2$: Fundamental Theorem of Arithmetic says that in that case, both $p$ and $p-1$ would be powers of $2$. Unless $p-1$=1, of course. For 2, note that not even the $9$-sided regular polygon is constructible.
– Lubin
Dec 3 '18 at 22:05
add a comment |
1
I edited your post to make the $LaTeX$ work. Remember, "$" and ${ cdot }$ are your two best friends in $LaTeX$. Cheers!
– Robert Lewis
Dec 2 '18 at 23:22
Did you show that $Phi_p(t) = sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $zeta_p$ ?
– reuns
Dec 3 '18 at 0:12
1. It can’t be that $p(p-1)$ is a power of $2$: Fundamental Theorem of Arithmetic says that in that case, both $p$ and $p-1$ would be powers of $2$. Unless $p-1$=1, of course. For 2, note that not even the $9$-sided regular polygon is constructible.
– Lubin
Dec 3 '18 at 22:05
1
1
I edited your post to make the $LaTeX$ work. Remember, "$" and ${ cdot }$ are your two best friends in $LaTeX$. Cheers!
– Robert Lewis
Dec 2 '18 at 23:22
I edited your post to make the $LaTeX$ work. Remember, "$" and ${ cdot }$ are your two best friends in $LaTeX$. Cheers!
– Robert Lewis
Dec 2 '18 at 23:22
Did you show that $Phi_p(t) = sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $zeta_p$ ?
– reuns
Dec 3 '18 at 0:12
Did you show that $Phi_p(t) = sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $zeta_p$ ?
– reuns
Dec 3 '18 at 0:12
1. It can’t be that $p(p-1)$ is a power of $2$: Fundamental Theorem of Arithmetic says that in that case, both $p$ and $p-1$ would be powers of $2$. Unless $p-1$=1, of course. For 2, note that not even the $9$-sided regular polygon is constructible.
– Lubin
Dec 3 '18 at 22:05
1. It can’t be that $p(p-1)$ is a power of $2$: Fundamental Theorem of Arithmetic says that in that case, both $p$ and $p-1$ would be powers of $2$. Unless $p-1$=1, of course. For 2, note that not even the $9$-sided regular polygon is constructible.
– Lubin
Dec 3 '18 at 22:05
add a comment |
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1
I edited your post to make the $LaTeX$ work. Remember, "$" and ${ cdot }$ are your two best friends in $LaTeX$. Cheers!
– Robert Lewis
Dec 2 '18 at 23:22
Did you show that $Phi_p(t) = sum_{n=0}^{p-1} t^n$ is the minimal polynomial of $zeta_p$ ?
– reuns
Dec 3 '18 at 0:12
1. It can’t be that $p(p-1)$ is a power of $2$: Fundamental Theorem of Arithmetic says that in that case, both $p$ and $p-1$ would be powers of $2$. Unless $p-1$=1, of course. For 2, note that not even the $9$-sided regular polygon is constructible.
– Lubin
Dec 3 '18 at 22:05