By Raymond Ayoub

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**Example text**

Rs ) with ri ∈ Ri ; the ring operations are defined as follows f1e1 (r1 , . . , rs ) + (r1′ , . . , rs′ ) = (r1 + r1′ , . . , rs + rs′ ) and (r1 , . . , rs )(r1′ , . . , rs′ ) = (r1 r1′ , . . , rs rs′ ). Then R is again a ring. We denote the ideal generated by f ∈ k[x] by f , so f = {gf | g ∈ k[x]}. Also, we write [h]f = h + f for the coset of h modulo f . If it is clear which f is meant, then we also write [h]. The polynomial h is called a representative of the coset [h]f ; note that a coset does not have a unique representative, indeed, all h + gf are representatives of the same coset.

9 has to appear exactly once. The sudoku is said to be well-posed if there is exactly one solution. The key observation is that a sudoku is an instance of a graph colouring problem. There are diverse methods to code such a problem into polynomial equations. Here we follow the exposition in a recent book of Decker and Pfister (A First Course in Computational Algebraic Geometry, Cambridge University Press 2013). 28 Gr¨obner Bases 3 5 9 8 2 7 3 1 3 4 1 5 2 8 7 5 6 3 2 1 9 6 4 8 2 9 7 3 7 5 In order get polynomials from a sudoku we first number the boxes from 1 to 81.

If k = R we can make a graph of the curve. It will look like this: 48 Integer Factorisation Note that the graph has to be symmetric with respect to the x-axis, as the equation is y 2 = f (x). Hence (x0 , y0 ) ∈ E(k) implies (x0 , −y0 ) ∈ E(k). One of the main points of elliptic curves is that we can define an addition + on it that makes it into an abelian group. , P + O = O + P = P for all P ∈ E(k). - If P1 , P2 ∈ E(k), P1 = P2 and P1 , P2 = O then we construct the line ℓ through P1 and P2 . The line ℓ intersects E(k) in a third point P3 = (x3 , y3 ), and we set P1 + P2 = (x3 , −y3 ).