By Timothy Gowers (ed)

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Pursuing this idea, one can calculate that the entry in row i and column j of the matrix P of f g is ai1 b1j + ai2 b2j + · · · + ain bnj . This matrix P is called the product of A and B and is written AB. If you have not seen this deﬁnition then you will ﬁnd it hard to grasp, but the main point to remember is that there is a way of calculating the matrix for f g from the matrices A, B of f and g, and that this matrix is denoted AB. Matrix multiplication of this kind is associative but not commutative.

It turns out that both choices are possible: one automorphism is the “trivial” √ √ one f (a + b 2) = a + b 2 and the other is the more √ √ interesting one f (a + b 2) = a − b 2. This observation demonstrates that there is no algebraic diﬀerence between the two square roots; in this sense, the ﬁeld √ Q( 2) does not know which square root of 2 is positive and which negative. 3. PUP: proofreader marked this up to be a full stop (without preceding space) and then the ellipsis but that seems wrong to me.

However, there is a more sophisticated view, ﬁrst PUP: proofreader wanted a comma here but Tim would strongly prefer not to insert one. OK to keep it as it is I presume? 3. 56], which regards transformations as the true subject matter of geometry. 1, transformations go hand in hand with groups, and for this reason there is an intimate connection between geometry and group theory. Indeed, given any group of transformations, there is a corresponding notion of geometry, in which one studies the phenomena that are unaﬀected by transformations in that group.