12. Let F = Z/pZ then show: (a) There exists f € F[t] with deg f = 2 and f irreducible. (b) Use the f in (a) to construct a field with p² elements. (c) If f₁, f2 € F[t] have deg fi fields are isomorphic. = 2 and fi irreducible for i = 1,2, show that their splitting

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Field theory

12. Let FZ/pZ then show:
(a) There exists f = F[t] with deg f = 2 and f irreducible.
(b) Use the f in (a) to construct a field with p² elements.
(c) If f₁, f2 € F[t] have deg fi = 2 and f; irreducible for i = 1, 2, show that their splitting
fields are isomorphic.
Transcribed Image Text:12. Let FZ/pZ then show: (a) There exists f = F[t] with deg f = 2 and f irreducible. (b) Use the f in (a) to construct a field with p² elements. (c) If f₁, f2 € F[t] have deg fi = 2 and f; irreducible for i = 1, 2, show that their splitting fields are isomorphic.
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