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second proof of Wedderburn's theorem
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(Proof)
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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
- $ \Phi_n(q)=\prod_{\xi \in P_n}(q- \xi)$
- $ p(q)=q^n-1=\prod _{\xi \in G_n} (q- \xi)=\prod_{d\mid n}\Phi_d(q) $
- $ \Phi_n(q)\in \znums [q] \;$ , it has multiplicative identity and $\Phi_n(q)\mid q^n-1$
- $ \Phi_n(q) \mid \frac{q^n-1 }{q^d-1} \;$ with $ d \mid n, d<n$
by conjugacy class formula, we have: $$q^n-1=q-1+\sum_x \frac{q^n-1}{q^{n_x}-1} $$ by last two previous properties, it results: $$ \Phi_n(q) \mid q^n-1 \;,\; \Phi_n(q) \mid \frac{q^n-1}{q^{n_x}-1} \Rightarrow \Phi_n(q) \mid q-1$$
because $\Phi_n(q)$ divides the left and each addend of $ \sum_x \frac{q^n-1}{q^{n_x}-1} $ of the right member of the conjugacy class formula.
By third property $$q>1 \;,\; \Phi_n(x)\in \znums[x] \Rightarrow \Phi_n(q)\in \znums \Rightarrow |\Phi_n(q)| \mid q-1 \Rightarrow |\Phi_n(q)|\leqslant q-1 $$
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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"second proof of Wedderburn's theorem" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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| Keywords: |
cyclotomic polynomial |
This object's parent.
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Cross-references: primitive root of unity, triangle inequality, theorem, right, divides, properties, cyclotomic polynomial, unity, primitive roots, roots of unity, proof, conjugacy class formula, Zsigmondy's theorem, Wedderburn's theorem
This is version 14 of second proof of Wedderburn's theorem, born on 2003-04-20, modified 2006-10-04.
Object id is 4198, canonical name is SecondProofOfWedderburnsTheorem.
Accessed 3204 times total.
Classification:
| AMS MSC: | 12E15 (Field theory and polynomials :: General field theory :: Skew fields, division rings) |
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Pending Errata and Addenda
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