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I have a question on some definitions in algebraic geometry.

I just started to read the book Field Arithmetic, by M.Fried and M.Jarden, but I cannot find the definitions they use.

Fix $K$ a field. What is a absolutely irreducible variety over $K$? What is a $K$-rational point of a variety?

Do you have a reference where i can read about these definitions?

Thanks!

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An absolutely irreducible variety is a variety which is irreducible over an algebraically closed extension of $K$.

A $K$-rational points is just a point of the variety with coordinates in $K$.

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  • $\begingroup$ The coordinates of a point in the variety are not always in $K$? $\endgroup$ – El.Gon.Zalo May 2 '18 at 15:30
  • $\begingroup$ Usually, they're in an algebraic closure of $K$ (for instance, in projective geometry, among all conics, a (real) circle is characterised by the fact that it passes through the points at infinity $[1:i:0]$ and $[1:-i:0]$, which are therefore known as the cyclic points. $\endgroup$ – Bernard May 2 '18 at 15:44

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