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Tutorials for Algebra-III

Abhay Soman

Worksheet Tutorial-6

Following exercises are taken from Section 13.6 of the book Abstract Algebra by Dummit and Foote.
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1.

Show the following.
  1. Suppose that \(\zeta_n\) and \(\zeta_m\) are primitive \(n\)-th and primitive \(m\)-th roots of unity, respectively. If \(\gcd(m,n)=1\) then, show that \(\zeta_m\zeta_n\) is a primitive \(mn\)-th root of unity, and that for a divisior \(d\) of \(n\text{,}\) \(\left(\zeta_n\right)^d\) is a primitive \(\tfrac{n}{d}\)-th root of unity.
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  2. If \(a=p^kb\text{,}\) where \(p\) is a prime and \(p\not\mid b\) then, there are precisely \(b\) distinct \(a\)-th roots of unity over a field of characteristic \(p\text{.}\)
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  3. Let \(n\) be an odd natural number. Show that if a field contains the \(n\)-th roots of unity then, it contains the \(2n\)-th roots of unity.
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2.

Show that there are only a finite number of roots of unity in any finite extension \(K\) of \(\Q\text{.}\)
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3.

Show that for an odd natural number \(n>1\text{,}\) show that \(\Phi_{2n}(x)=\Phi(-x)\text{.}\)
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4.

Let \(A\in M_n(\C)\) be such that \(A^m=I_n\) for some \(m\in\N\text{.}\) Show that \(A\) is diagonalizable.
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Let \(F\) be a field of characteristic \(p\neq 0\text{,}\) and let \(A=\left(\begin{smallmatrix}1\amp a\\0\amp 1\end{smallmatrix}\right)\in M_2(F)\) be such that \(A^p=I_2\text{.}\) Show that if \(a\neq 0\) then, \(A\) is not diagonalizable.
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5.

Consider the Frobenious homomorphism \(\varphi\colon\F_{p^n}\to\F_{p^n}\) given by \(x\mapsto x^p\text{.}\)
  1. Show that \(\varphi\) is an isomorphism. Further show that \(\varphi^n=\unit_{\F_{p^n}}\) and that no lower power of \(\varphi\) is the identity.
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  2. Determine the rational canonical form of \(\varphi\) over \(\F_p\text{,}\) where \(\varphi\) is considered to be an \(\F_p\)-linear map on the \(n\)-dimensional vector space \(\F_{p^n}\) over \(\F_p\text{.}\)
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  3. Determine the Jordan canonical form over a field containing all the eigenvalues of \(\varphi\text{,}\) where \(\varphi\) is considered to be an \(\F_p\)-linear map on the \(n\)-dimensional vector space \(\F_{p^n}\) over \(\F_p\text{.}\)
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