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I am trying to show that if a line given by $y = mx + b$ intersects an Elliptic Curve given by $E(\mathbb{K}): y^2 = x^3 + Ax + B$ in three points then the line is not tangent to the curve.

Given that char$(\mathbb{K}) \neq 2,3$ and $\mathbb{K}$ is algebraically closed.

Also that if they intersect in two points, the line is tangent to the curve. And if they intersect in one point, the intersection is an inflection point.

I have tried to characterize the points of intersections and compare the slope of the line and curve at those points but I'm not getting anywhere.

any help is deeply appreciated.

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    It would help if you told us what definition of tangency you are using.2012-02-13
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    So basically using this equality $(mx +b)^2 = x^3 + Ax +B$ I tried to characterize the three points of intersection i.e. $(mx +b)^2 - x^3 - Ax -B = (x - x_1)(x - x_2)(x - x_3)$. But I'm not sure where to go from here.2012-02-13
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    Have you read my comment?2012-02-13
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    Sorry Mariano, I was trying to use the fact that $m$ is the slope of the line and try to somehow compare it with the derivative of the curve at those three points of intersections, i.e. $x_1, x_2, x_3$ I don't think this is a good approach as I have been thinking about it for some times. Do you have any other approach in mind?2012-02-13
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    Do you know what the characteristic of a field is?2012-02-13
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    char$(\mathbb{K})≠2,3$ and $\mathbb{K}$ is algebraically closed2012-02-13

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