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Contents Embed Dark Mode Prev Up Next \(\newcommand{\N}{\mathbb N}
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Worksheet Tutorial-2
Following exercises are taken from Section 13.2 of the book
Abstract Algebra by Dummit and Foote.
1. Let
\(\F\) be a finite field of characteristic
\(p\text{.}\) Show that the number of elements in
\(\F\) is
\(p^n\) for some natural number
\(n\text{.}\)
2. Suppose that
\(g(x)=x^2+x+1\) and
\(h(x)=x^3-x+1\text{.}\) Obtain fields of cardinality
\(4,8,9\) and
\(27\) by adjoining roots of either
\(g(x)\) or
\(h(x)\) to
\(\F_2\) or
\(\F_3\text{.}\)
3. Find the minimal polynomial of
\(1+i\in\C\) over
\(\Q\text{.}\)
4.
Determine the following degrees.
\begin{equation*}
\left[\Q(2+\sqrt{3}):\Q\right],\quad\left[\Q\left(\sqrt{3+2\sqrt{2}}\right):\Q\right],\quad\text{and}\quad\left[\Q\left(1+\sqrt[3]{2}+\sqrt[3]{4}\right):\Q\right]
\end{equation*}
5. Show that
\(x^3-2\) and
\(x^3-3\) are irreducible over
\(\Q(i)\text{.}\)
