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Last Lesson:. Upshot: We can show a system of equations has no solutions by computing the reduced Groebner basis. . Application: Radical Membership. Question: Given an ideal , can we find generators for its radical, ?.

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Last Lesson:

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  1. Last Lesson: Upshot: We can show a system of equations has no solutions by computing the reduced Groebner basis.

  2. Application: Radical Membership Question: Given an ideal , can we find generators for its radical, ? While existing algorithms are impractical, we can determine radical membership... Proposition (Radical Membership) Let k be an arbitrary field and let be an ideal. Then iff

  3. Radical Membership proof... Observe (as in the proof of Hilbert’s Nullstellensatz): If ,then we can write 1 as a polynomial combination: Setting yields Multiplying by high enough power, , we get

  4. Radical Membership proof... We have established the forward implication: If then . For the reverse implication, if then for some positive integer m. Next we get sneaky: both in I Hence

  5. Radical Membership Test We have established: iff . The Consistency Theorem then tells us we can test if by computing the reduced Groebner basis of If the reduced Groebner basis of is {1} then .

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