Speaking back and forth with math professor about convoluted bullcrap which doesn't matter.
⚗️alchemist⚗️ (npub1pt6…6mf6) say we have F is a field, K is an integral domain, F \subseteq K. Let N_\alpha be the ideal in F[x] which contains all polynomials in F[x] with \alpha as a root. Let \alpha be in K - F (and therefore (x - \alpha) \in K[x] - F[x]))
Because F[x] is a PID, N_\alpha must have a unique generator, which because (x - \alpha) \notin F[x], must be of at least degree 2. So, there must exist some g(x) \in K[x] (possible g(x) = 0) such that (x - \alpha)g(x). However, I don't see *why* g(x) must be unique. Why can't there exist two polynomials g(x), h(x) in K[x] such that gcd(g(x)(x-\alpha), h(x)(x-\alpha)) restricted to F[x] = 1, and therefore N_\alpha = (g(x)(x-\alpha), h(x)(x-\alpha))?
I figure it must be the case that h(x) = kg(x), k \in K, because F[x] is a PID, but I'm missing something here.
Guy Incognito /
npub1pz…j7c8z
2023-03-28 23:55:49
Author Public Key
npub1pz6u0prt75nvmyfhgpngncrvq7xf7cs8f3xgnfd79j7uknj978mstj7c8zPublished at
2023-03-28 23:55:49Event JSON
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"content": "Speaking back and forth with math professor about convoluted bullcrap which doesn't matter.\n\n#[0] say we have F is a field, K is an integral domain, F \\subseteq K. Let N_\\alpha be the ideal in F[x] which contains all polynomials in F[x] with \\alpha as a root. Let \\alpha be in K - F (and therefore (x - \\alpha) \\in K[x] - F[x]))\n\nBecause F[x] is a PID, N_\\alpha must have a unique generator, which because (x - \\alpha) \\notin F[x], must be of at least degree 2. So, there must exist some g(x) \\in K[x] (possible g(x) = 0) such that (x - \\alpha)g(x). However, I don't see *why* g(x) must be unique. Why can't there exist two polynomials g(x), h(x) in K[x] such that gcd(g(x)(x-\\alpha), h(x)(x-\\alpha)) restricted to F[x] = 1, and therefore N_\\alpha = (g(x)(x-\\alpha), h(x)(x-\\alpha))?\n\nI figure it must be the case that h(x) = kg(x), k \\in K, because F[x] is a PID, but I'm missing something here.",
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