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@@ -734,11 +734,107 @@
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"source": [
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"As observed we have a uniformizer $t$ with $k = 2$."
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]
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+ },
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+ {
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+ "cell_type": "markdown",
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+ "id": "a05bc126",
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+ "metadata": {},
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+ "source": [
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+ "# Valuation with Places\n",
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+ "\n",
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+ "Using sage valuation rings since local ring is a DVR"
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+ ]
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+ },
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+ {
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+ "cell_type": "code",
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+ "execution_count": 2,
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+ "id": "3da404fc",
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+ "metadata": {},
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+ "outputs": [
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+ {
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+ "data": {
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+ "text/plain": [
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+ "[Place (x, y),\n",
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+ " Place (x + 3, y + 4),\n",
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+ " Place (x + 3, y + 7),\n",
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+ " Place (x + 5, y + 3),\n",
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+ " Place (x + 5, y + 8),\n",
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+ " Place (x + 7, y + 5),\n",
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+ " Place (x + 7, y + 6),\n",
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+ " Place (x + 9, y + 4),\n",
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+ " Place (x + 9, y + 7),\n",
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+ " Place (x + 10, y + 4),\n",
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+ " Place (x + 10, y + 7)]"
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+ ]
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+ },
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+ "execution_count": 2,
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+ "metadata": {},
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+ "output_type": "execute_result"
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+ }
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+ ],
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+ "source": [
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+ "K.<x> = FunctionField(GF(11))\n",
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+ "_.<Y> = K[]\n",
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+ "L.<y> = K.extension(Y^2 - x^3 - 4*x)\n",
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+ "L.places_finite()"
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+ ]
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+ },
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+ {
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+ "cell_type": "markdown",
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+ "id": "e4719879",
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+ "metadata": {},
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+ "source": [
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+ "We are interested in $P = (2, 4)$ so we want the place $(x - 2, x - 4) = (x + 9, y + 7)$."
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+ ]
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+ },
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+ {
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+ "cell_type": "code",
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+ "execution_count": 3,
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+ "id": "f87c8b23",
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+ "metadata": {},
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+ "outputs": [
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+ {
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+ "data": {
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+ "text/plain": [
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+ "Place (x + 9, y + 7)"
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+ ]
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+ },
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+ "execution_count": 3,
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+ "metadata": {},
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+ "output_type": "execute_result"
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+ }
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+ ],
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+ "source": [
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+ "# I don't know how to actually construct this yet\n",
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+ "p = L.places_finite()[-3]\n",
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+ "p"
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+ ]
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+ },
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+ {
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+ "cell_type": "code",
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+ "execution_count": 4,
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+ "id": "3d007b0b",
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+ "metadata": {},
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+ "outputs": [
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+ {
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+ "data": {
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+ "text/plain": [
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+ "2"
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+ ]
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+ },
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+ "execution_count": 4,
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+ "metadata": {},
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+ "output_type": "execute_result"
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+ }
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+ ],
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+ "source": [
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+ "(y - 2*x).valuation(p)"
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+ ]
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}
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],
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"metadata": {
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"kernelspec": {
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- "display_name": "SageMath 9.6",
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+ "display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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@@ -752,7 +848,7 @@
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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- "version": "3.10.5"
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+ "version": "3.10.7"
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}
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},
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"nbformat": 4,
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