I am read Abstrac Álgebra, Gallian and I am thinking how to prove this Theorem The first implications is ready but the second implication I dont know... Can you give me a Idea. Let F be a field, I a nonzero ideal in F[x], and g(x) an element of F[x]. Then, I=⟨g(x)⟩ if and only if g(x) is a nonzero polynomial of minimum degree in I.
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What is the second implication for you? – Bernard Oct 20 '19 at 23:22
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if g(x) is a nonzero polynomial of minimum degree in I Then, I=⟨g(x)⟩ – wessi Oct 20 '19 at 23:39
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Hint:
If what you call the second implication is having a minimal degree ensures we have generator for $I$, for any polynomial $f(x)$, perform a euclidean division by $g(x)$. What can you say of the remainder $r(x)$?
Bernard
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