Dave Rusin <email@example.com> [sci.math 3 Dec 1999 21:49:12 GMT]
>The following example was pointed out to me by John Wolfskill. > >(1) Show that sqrt( 8 + 3 sqrt(7) ) is the sum of the square > roots of > two integers. > >(2) Generalize. > >This came up as an illustration of some topics in Galois >theory -- what looks like it ought to involve a dihedral >Galois group lies in a field with group Z/2 x Z/2 -- but >the topic expands a bit upon generalization, e.g., just >which two quadratic extensions of Q have a compositum big >enough to hold the required algebraic numbers. > >dave
You might want be interested in
Webster Wells, "Advanced Course in Algebra", D. C. Heath and Co., 1904 [pp. 235-237]
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ARTICLE 390: If sqrt[a + sqrt(b)] = sqrt(x) + sqrt(y), where a, b, x, and y are rational expressions, and a greater than sqrt(b), then sqrt[a - sqrt(b)] = sqrt(x) - sqrt(y).
ARTICLE 391: The preceding principles may be used to find the square root of certain expressions which are in the form of the sum of a rational expression and a quadratic surd.
Then by #390, sqrt[13 + sqrt(160)] = sqrt(x) + sqrt(y).
Multiplying the equations gives sqrt(169 - 160) = x - y. Hence, x - y = 3. Squaring (1), 13 - sqrt(160) = x - 2*sqrt(xy) + y. Hence, x + y = 13. [Citing the previously proved result that a + sqrt(b) = c + sqrt(d), where a, c are rational and b, d are quadratic surds, implies a=c and b=d.] It now easily follows that x=8 and y=5.
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Using this method for sqrt[8 + 3*sqrt(7)] gives
sqrt[8 + 3*sqrt(7)] = sqrt(9/2) + sqrt(7/2).
You can't write sqrt[8 + 3*sqrt(7)] as a sum sqrt(x) + sqrt(y) with x and y both integers, however. [There are not many pairs of integers to check. For instance, among other restrictions, 1 <= x,y <= 8, since the number in question is less than 4.]
Wells goes on to show that
sqrt[8 + sqrt(48)] = sqrt(6) + sqrt(2)
sqrt[22 - 3*sqrt(32)] = 3*sqrt(2) - 2
sqrt(392) + sqrt(360) = [2^(1/4)]*[3 + sqrt(5)]
ARTICLE 394: It may be proved, as in #390, that if cube:root[a + sqrt(b)] = x + sqrt(y), where a, x are rational expressions and sqrt(b), sqrt(y) quadratic surds, then cube:root[a - sqrt(b)] = x - sqrt(y).