??so consider, for example, say, Z[x]/(x^37-1)/(x-1) ...

???prime ideals here ... ????..... ???mapping down to prime ideals of Z ...

???should be pretty clear what these prime ideals are, in general ... ????....

???but then what about principalness of such ideals ???? ....

??compare to examples that we've thought more about ... ??? .....

?? prime p with 37 dividing p-1 ... ???? p = 1 mod 37 ....

1 38 75 112 149 186 223 260 297 334 371 408

149 223 ...

Z[x]/(x^37-1)/(x-1) -> Z/149

1 2 4 8 16 32 64 128 107 65 130 111 73 146 143 137 125 101 53 106 63 126 103 57 114 79 9 18 36 72 144 139 129 109 69 138 127 105 61 122 95 41 82 15 30 60 120 91
33 66 132 115 81 13 26 52 104 59 118 87 25 50 100 51 102 55 110 71 142 135 121 93 37 74 148 147 145 141 133 117 85 21 42 84 19 38 76 3 6 12 24 48 96 43 86 23 46 92 35 70 140 131 113 77 5 10 20 40 80 11 22 44 88 27 54 108 67 134 119 89 29 58 116 83 17 34 68 136 123 97 45 90 31 62 124 99 49 98 47 94 39 78 7 14 28 56 112 75


x |-> 16

x |-> 107

x |-> 73

x |-> 125

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x |-> 28

???so for example the ideal given as kernel of x |-> 16 here ... ??? what are some ways of thinking abut whether it's principal ?? .... ??? ...

?? well, perhaps there's a naive direct way ... ???

x-16 is in the kernel ...

x^2-107 is as well ...

x^3-73

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hmm, well, also 149 is in the kernel, right ?? ....

??well, let's try testing how close x-16 comes to generating the ideal ...

so, we're setting x to 16 ... ?so basically we're taking Z and forcing (16^37-1)/(16-1) to be zero ?? .... hmmm ...

??? by the way i'm assuming for the moment that the algebraic integers here are the obvious ones ... i think todd said that that's what he thought they were in this case ...