Trying to understand why the zeta function is a rational function under certain conditions. Questions about...












4














Information: I linked the pages below, which relate to my questions.



I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:



$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$



That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$



$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$



That was my first question. Next:



$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$



in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:



Now they use Proposition 11.1.1 to get:



$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.



I'm aware that you need context to answer my questions. So here are the pages:



enter image description hereenter image description here



And here is Proposition 11.1.1:



enter image description here



If you need something more, let me know.
Thank you for your help.






number-theory finite-fields characters zeta-functions







share|cite|improve this question






















  • $n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
    – reuns
    Nov 25 at 1:17


















4














Information: I linked the pages below, which relate to my questions.



I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:



$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$



That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$



$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$



That was my first question. Next:



$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$



in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:



Now they use Proposition 11.1.1 to get:



$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.



I'm aware that you need context to answer my questions. So here are the pages:



enter image description hereenter image description here



And here is Proposition 11.1.1:



enter image description here



If you need something more, let me know.
Thank you for your help.






number-theory finite-fields characters zeta-functions







share|cite|improve this question






















  • $n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
    – reuns
    Nov 25 at 1:17
















4












4








4


2





Information: I linked the pages below, which relate to my questions.



I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:



$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$



That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$



$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$



That was my first question. Next:



$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$



in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:



Now they use Proposition 11.1.1 to get:



$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.



I'm aware that you need context to answer my questions. So here are the pages:



enter image description hereenter image description here



And here is Proposition 11.1.1:



enter image description here



If you need something more, let me know.
Thank you for your help.






number-theory finite-fields characters zeta-functions







share|cite|improve this question













Information: I linked the pages below, which relate to my questions.



I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:



$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$



That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$



$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$



That was my first question. Next:



$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$



in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:



Now they use Proposition 11.1.1 to get:



$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.



I'm aware that you need context to answer my questions. So here are the pages:



enter image description hereenter image description here



And here is Proposition 11.1.1:



enter image description here



If you need something more, let me know.
Thank you for your help.






number-theory finite-fields characters zeta-functions




number-theory finite-fields characters zeta-functions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 24 at 12:15









RukiaKuchiki

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299211












  • $n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
    – reuns
    Nov 25 at 1:17




















  • $n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
    – reuns
    Nov 25 at 1:17


















$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17






$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17

















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