It’s maybe worth saying that while you can get √2 in, say, Q7, (1) this isn’t really “the same” √2 as you have in the reals—I mean, it is a square root of 2, just as the corresponding thing in the reals is, but the structure it fits into isn’t the same as that of the reals—and (2) just as some square roots (those of negative numbers) don’t exist in R, so some square roots (but a different set) don’t exist in any given Qp. For instance, there is no square root of 2 in Q2 or Q3 or in Q5, which is why I said Q7 before.
(In Qp, just as in R, in some sense “about half” of integers have square roots.)
[EDITED to add:] Actually I see that Viliam already noticed that not everything has a square root in the p-adics.
A couple more remarks. The things Viliam constructed are (aside from the base-10 versus base-p thing) the p-adic integers, generally written Zp; the full p-adic field Qp is what you get if you allow finitely many nonzero digits after the “decimal” point, just as when writing ordinary numbers we allow finitely many nonzero digits before the decimal point.
To get all the square roots (and much more), you can construct an “algebraic closure” of Qp just as you can for R. But the story here isn’t quite as nice as it is when you go from R to C.
You get R from Q by doing a “topological completion”, and then C from R by doing an “algebraic closure”, but then C is still topologically complete. Similarly, you get Qp from Q by doing a topological completion (using a different topology), and then you can construct an algebraic closure of that … but then the result isn’t topologically complete any more.
There is a simple explicit construction to get from R to C: you throw in a square root of −1 and then every other polynomial equation becomes solvable. (Which is the definition of being “algebraically closed”.) That isn’t the case for Qp, whose algebraic closure is not a nice “finite extension” like that. If you extend Q2 or Q5 just enough to get a square root of 2, that doesn’t automatically get you all the other square roots, or solutions to all the other polynomial equations that don’t have ’em.
An other funny thing you can do with square roots: let’s take Q7, and let us look at the power series √1+X=1+12X−18X⋯. This converges for |X|<1, so that you can specialize in X=79. Now, you can also do that inside R, and the series converges to 34, ``the″ square root of 916. But in Q7 this actually converges to −34.
A further remark: confusingly, the algebraic closure of Qp and its completion Cp are actually isomorphic as fields (and both are also isomorphic to C), since they are both algebraically closed fields of characteristic zero and of cardinality of the continuum.
It’s maybe worth saying that while you can get √2 in, say, Q7, (1) this isn’t really “the same” √2 as you have in the reals—I mean, it is a square root of 2, just as the corresponding thing in the reals is, but the structure it fits into isn’t the same as that of the reals—and (2) just as some square roots (those of negative numbers) don’t exist in R, so some square roots (but a different set) don’t exist in any given Qp. For instance, there is no square root of 2 in Q2 or Q3 or in Q5, which is why I said Q7 before.
(In Qp, just as in R, in some sense “about half” of integers have square roots.)
[EDITED to add:] Actually I see that Viliam already noticed that not everything has a square root in the p-adics.
A couple more remarks. The things Viliam constructed are (aside from the base-10 versus base-p thing) the p-adic integers, generally written Zp; the full p-adic field Qp is what you get if you allow finitely many nonzero digits after the “decimal” point, just as when writing ordinary numbers we allow finitely many nonzero digits before the decimal point.
To get all the square roots (and much more), you can construct an “algebraic closure” of Qp just as you can for R. But the story here isn’t quite as nice as it is when you go from R to C.
You get R from Q by doing a “topological completion”, and then C from R by doing an “algebraic closure”, but then C is still topologically complete. Similarly, you get Qp from Q by doing a topological completion (using a different topology), and then you can construct an algebraic closure of that … but then the result isn’t topologically complete any more.
There is a simple explicit construction to get from R to C: you throw in a square root of −1 and then every other polynomial equation becomes solvable. (Which is the definition of being “algebraically closed”.) That isn’t the case for Qp, whose algebraic closure is not a nice “finite extension” like that. If you extend Q2 or Q5 just enough to get a square root of 2, that doesn’t automatically get you all the other square roots, or solutions to all the other polynomial equations that don’t have ’em.
An other funny thing you can do with square roots: let’s take Q7, and let us look at the power series √1+X=1+12X−18X⋯. This converges for |X|<1, so that you can specialize in X=79. Now, you can also do that inside R, and the series converges to 34, ``the″ square root of 916. But in Q7 this actually converges to −34.
A further remark: confusingly, the algebraic closure of Qp and its completion Cp are actually isomorphic as fields (and both are also isomorphic to C), since they are both algebraically closed fields of characteristic zero and of cardinality of the continuum.
Yup. Very different topologically, though.