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Ideals and GCDs in k[x] .... .... Cox et al ....
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[QUOTE="fresh_42, post: 5511575, member: 572553"] The easy part is ##I := <f_1, \dots , f_s> \;⊆ \;<h>\; ⊆\; <h,f_1>\;## because every ##f_i## can be written ##f_i = g_i \cdot h## by definition of ##h##. Thus any ##f = \sum a_i f_i ∈ I## gets ##f = \sum a_i g_i h ∈ \; <h>.## The other way, your first, is essentially Bézout's Lemma. I'm too lazy to type it from Wikipedia. Maybe you get along with what is written there: [URL]https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity[/URL] It might as well be the quoted proposition 6 or corollary 4. It means that you can write ##h_2 = p_1 f_1 + p_2 f_2## if ##h_2## is the greatest common divisor of ##f_1## and ##f_2## and then by induction ##h = p_1 f_1 + \dots + p_s f_s## which is what we need for ##h \in I.## It applies to PID, so you may substitute "integers" by "polynomials". There is a written proof and a link to the polynomial version (without proof as far as I could see): [URL]https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#B.C3.A9zout.27s_identity_and_extended_GCD_algorithm[/URL] If you have further questions to it or about the Wiki entry, please let me know. [/QUOTE]
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Ideals and GCDs in k[x] .... .... Cox et al ....
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