1.
Find all irreducible polynomials of degree 1, 2 and 4 over \(\F_2\text{,}\) and show that their product is \(x^{16}-x\text{.}\)
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Worksheet Tutorial-5
Following exercises are taken from Section 13.5 of the book Abstract Algebra by Dummit and Foote.
2.
Show that \(d\) divides \(n\) if and only if \(x^d - 1\) divides \(x^n - 1\text{.}\) Further show that \(\F_{p^d}\subseteq\F_{p^n}\) if and only if \(d\) divides \(n\text{.}\)
3.
Let \(p\) be a prime and let \(0\neq a\in\F_p\text{.}\) Show that \(x^p - x + a\) is irreducible and separable over \(\F_p\text{.}\)
4.
Suppose that \(K\) is a field and that it is not a perfect field. Prove that there exist irreducible inseparable polynomials over \(K\text{.}\) Conclude that there exist inseparable finite extensions of \(K\text{.}\)
5.
Let \(K/F\) be field extension, and let \(F\) be a perfect field. Suppose that \(f(x)\in F[x]\) has no repeated irreducible factors in \(F[x]\text{.}\) Show that \(f(x)\) has no repeated irreducible factors in \(K[x]\text{.}\)
