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25-05-2023
Mathematics

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3. Let F = Z5 and let f(x) = x³ + 2x + 1 € F[r]. Let a be a root of f(x) in some extension of F. (a) Show that f(x) is irreducible in F[2]. (b) Find [F(a): F] and find a basis for F(a) over F. How many elements does F(a) have? (c) Write a + 2a + 3 in the form co + cia + c₂a².

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