Question

(a) Prove that p(x) = x3 + 9x2 + 18x + 13 is irreducible over Q. (b) Factorise 224 – 1 into monic irreducibles over Q, explaining carefully why each factor is irreducible. (c) Identify all possible isomorphisms between the following quotient rings: F5[x]/(22+x+1), F5[x]/(x² +2+2), F5[2]/(x2 + x + 3).

Please answer (c)!

Answer 1

O f(x) has degree 2 or 3 is irreducible over a field if if and only if feu) has a root in f. if f(x) in imeducible then REW) is field. over R <fery> Risring order same ② Any two field of ilomorphic. F [n] 11 f(H) = 1 52 > 5x2+x+1) fa (x) f(2)= 2 +13)=3 flo) = 1 f(1) = 3 so f does not have root in its fs [u] is field of order <x2++1) g(0) = 2 gel) = 4 12) = 3 <x²+2+2) 8 g(3)=4 H14)-2 no not in its So is field of order 5.2 <H424+2) 2 (n) has

F F [w] h(0)=3 AU)=0 <x'+x+3) 11 h(x) hind has noot es in its lu] IFs [n] is not field. So (x2+x+3) So As [n) Flu] isomorphic <x+14+13 <x²+x+27
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