# second proof of Wedderburn's theorem

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\newcommand {\cnums}{\mathbb{C}}
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We can prove Wedderburn's theorem,without using Zsigmondy's theorem on the conjugacy class formula of the first proof;
let $G_n$ set of n-th roots of unity and $P_n$ set of n-th primitive
roots of unity and $\Phi_d(q)$ the d-th cyclotomic polynomial.\\
It results
\begin{itemize}
\item $\Phi_n(q)=\prod_{\xi \in P_n}(q- \xi)$
\item $p(q)=q^n-1=\prod _{\xi \in G_n} (q- \xi)=\prod_{d\mid n}\Phi_d(q)$
\item $\Phi_n(q)\in \znums [q] \;$, it has multiplicative identity and $\Phi_n(q)\mid q^n-1$
\item $\Phi_n(q) \mid \frac{q^n-1 }{q^d-1} \;$with $d \mid n, d1 \;,\; \Phi_n(x)\in \znums[x] \Rightarrow \Phi_n(q)\in \znums \Rightarrow |\Phi_n(q)| \mid q-1 \Rightarrow |\Phi_n(q)|\leqslant q-1 $$\\If, for n>1,we have |\Phi_n(q)|>q-1 , then n=1 and the theorem is proved. \\We know that \\$$ |\Phi_n(q)|=\prod_{\xi \in P_n} |q - \xi|\;,\;with\; q- \xi\in \cnums $$\\by the triangle inequality in \cnums$$ |q-\xi|\geqslant||q|-|\xi||=|q-1|$$as \xi is a primitive root of unity, besides$$|q-\xi|=|q-1| \Leftrightarrow \xi=1$$but$$n>1 \Rightarrow \xi \neq 1$$therefore, we have$$|q-\xi|>|q-1|=q-1 \Rightarrow |\Phi_n(q)|>q-1$\$
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