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This tag is used if a reference is needed in a paper or textbook on a specific result.

18 votes
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Number of primitive $n$th roots with positive versus negative real parts

Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ …
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8 votes

Generalization of Schur's Lemma

Schur's lemma has a different generalization when the coefficient field $F$ is not algebraically closed. Then you get $M_{m_1}(D_1)\times\cdots\times M_{m_k}(D_k)$ where $D_i:={\rm Hom}_{FG}(\rho_i,\r …
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2 votes

Reference for restriction of a simple module over a splitting field to a smaller field?

One uses a matrix version of Hilbert's Satz 90 to answer Geoff's comment "If the field $E$ is not this minimal field, it seems less obvious to me how to realise the representation over the subfield ge …
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1 vote

How to detect if a subgroup lands inside an orthogonal group?

This answer may be more general than you need. Let $F$ be a field, and let $G=\langle g_1,\dots,g_m\rangle$ be a subgroup of ${\rm GL}(n,F)$ which preserves a form, with matrix $J$. Then $g_i J g_i^{\ …
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0 votes

Number of primitive $n$th roots with positive versus negative real parts

There are (at least) two explicit formulas. First, $D(n)$ can be shown to be a multiplicative function (this means $\gcd(m,n)=1$ implies $D(mn)=D(m)D(n)$), and the values of $D(p^k)$ for all primes $p …
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