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Worksheet Tutorial-10
Following exercises are taken from Section 14.3 of the book Abstract Algebra by Dummit and Foote.
2.
Show that an algebraically closed field must be infinite.
3.
Construct the finite field with \(16\) elements as a quotient of \(\F_2[x]\) and find a generator for its multiplicative group.
4.
Find an explicit isomorphism between the splitting fields of \(x^3-x+1\) and \(x^3-x-1\) over \(\F_3\text{.}\)
5.
Let \(K=\Q\left(\sqrt{D_1},\sqrt{D_2}\right)\) be a biquadratic extension over \(\Q\text{,}\) and let \(\theta\in K\) be such that \(K=\Q(\theta)\text{.}\) Show that the minimal polynomial of \(\theta\) over \(\Q\) is irreducible of degree \(4\) but is reducible modulo \(p\) for every prime \(p\text{.}\)
6.
Show that the splitting field of the polynomial \(x^p-x-a\) over \(\F_p\text{,}\) for \(0\neq a\in\F_p\) is cyclic.
7.
Let \(p\) be a prime number and let \(q=p^n\text{.}\) Denote by \(\sigma_q\) the \(m\)-th power of the Frobenius automorphism.
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Show that every finite extension of \(\F_q\) is Galois with cyclic Galois group generated by \(\sigma_q\text{.}\)
