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Tutorials for Algebra-III

Abhay Soman

Worksheet Tutorial-10

Following exercises are taken from Section 14.3 of the book Abstract Algebra by Dummit and Foote.
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3.

Construct the finite field with \(16\) elements as a quotient of \(\F_2[x]\) and find a generator for its multiplicative group.
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4.

Find an explicit isomorphism between the splitting fields of \(x^3-x+1\) and \(x^3-x-1\) over \(\F_3\text{.}\)
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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 \(p\text{.}\)
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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.
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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.
  1. Show that \(\sigma_q\) fixes \(\F_q\text{.}\)
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  2. Show that every finite extension of \(\F_q\) is Galois with cyclic Galois group generated by \(\sigma_q\text{.}\)
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  3. Prove that \(n\) divides \(\varphi(q-1)\text{.}\)
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8.

Prove that \(x^{p^n}-x+1\) is irreducible over \(\F_p\) only when \(n=1\) or \(n=p=2\text{.}\)
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