Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let...

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Prove the theorem in the lecture:Euclidean Domains and UFD's
Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is amaximal ideal in F[x] if and only if p(x) is irreducible overF.

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Denote IThen we have I is principal I for some px Fx 1 Assume px is irreducible and let I1 be an ideal of Fx containing I Then I1 for some fx Fx Since px I1 See Answer

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Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let...

50.1K

Verified Solution

Question

Advance Math

Prove the theorem in the lecture:Euclidean Domains and UFD's
Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is amaximal ideal in F[x] if and only if p(x) is irreducible overF.

Answer & Explanation Solved by verified expert

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Prove the theorem in the lecture:Euclidean Domains and UFD's Let F be a field, and let...

50.1K

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Advance Math

Prove the theorem in the lecture:Euclidean Domains and UFD'sLet F be a field, and let p(x) in F[x]. Prove that (p(x)) is amaximal ideal in F[x] if and only if p(x) is irreducible overF.

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Prove the theorem in the lecture:Euclidean Domains and UFD's
Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is amaximal ideal in F[x] if and only if p(x) is irreducible overF.