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The aim is to get from $1$ to $\sqrt[4]{2}$ or prove it is impossible using only one of the following options:

  1. Add or subtract two previously constructed numbers.
  2. Multiply two previously constructed numbers.
  3. Using a previously constructed number $\alpha$ construct both solutions to $\alpha^2+1=\beta^2$. I’ve managed to construct many numbers close to it, such as $\sqrt{4+2\sqrt{2}}$.
    I’m pretty sure it’s impossible but haven’t managed to prove it.
    Can any of you help?
    Edit 1: reciprocals
    We can construct all quadratic radicals and rational numbers.
    In this case, we are seaching for an extension of $\mathbb Q$ that is closed under (3), The constructible numbers are closed under this operation, but i think there is a subfield of the constructible numbers closed under it and containing $\mathbb Q$ as a subfield.
razivo
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    you are likely correct. Hartshorne gives $ \sqrt{ 1 + \sqrt 2 }$ as an impossible number. Likely to need Galois Theory to prove impossibility https://en.wikipedia.org/wiki/Pythagorean_field – Will Jagy Nov 28 '20 at 16:36
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    Norm considerations might be useful here. Note that the $\beta$ you construct will always have norm strictly larger than the $\alpha$ you use to construct it. $\alpha = -1,0,1$ construct $\beta = \sqrt{2},1,\sqrt{2}$ respectively. Every other constructible number using steps (1) and (2) (and thus (3)) has norm larger than $\sqrt{2}$. But your number has norm smaller than $\sqrt{2}$. – Pranav Chinmay Nov 28 '20 at 16:42
  • Explaining why this might be important could help to interest others. Until then: good luck with your problem. –  Nov 28 '20 at 16:43
  • How are the norm considerations effected by the ability to add and subtract constructed numbers? – razivo Nov 28 '20 at 16:49
  • not clear to me that your game allows reciprocals. Suggest you see if anything changes when allowing reciprocals and division. – Will Jagy Nov 28 '20 at 17:29
  • $\sqrt{4-2\sqrt{2}}$ is constructable but its norm/absolute value is smaller than $\sqrt[4]{2}.$ But there might be some other norm where this is true. – dezdichado Nov 28 '20 at 18:07
  • @WillJagy, edited to address your suggestion, it is possible using reciprocals. – razivo Nov 28 '20 at 18:13
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    @WillJagy, It was a problem posed by a friend of mine, I'm unsure if he actually solved it. – razivo Nov 28 '20 at 18:41
  • razivo your final calculation is in error, $\sqrt{4 - 2 \sqrt 2 }$ divided by $\sqrt 2$ is $\sqrt{2 - \sqrt 2 }$ – Will Jagy Nov 29 '20 at 02:48
  • @WillJagy, I've tried again and haven't found any, so the use of galois theory is probably necessary. – razivo Nov 29 '20 at 06:47

1 Answers1

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Alright, you should check your calculation again. If you really did construct $2^{1/4}$ you would immediately be able to construct $\sqrt{1 + \sqrt 2}.$ This is not possible: the quickest way to say it is that Hilbert's field is the set of totally real elements in the constructible field (closed under square roots of positive elements).

This is pages 145-148 in Geometry: Euclid and Beyond by Robin Hartshorne.

I repeated the first example search at https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/totallyreal_rel.html

and got

jagy@phobeusjunior:~$ sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath Version 6.9, Release Date: 2015-10-10                     │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage:  ZZx = ZZ['x']
sage:  F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 2, 10000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + (t + 1)*xF + 3*t - 3],
 [2048, x^4 - 4*x^2 + 2, xF^2 + t - 2],
 [2304, x^4 - 4*x^2 + 1, xF^2 + t*xF - 1],
 [2624, x^4 - 2*x^3 - 3*x^2 + 2*x + 1, xF^2 + (t + 1)*xF + t - 1],
 [4352, x^4 - 6*x^2 - 4*x + 2, xF^2 + t*xF + t - 2],
 [7168, x^4 - 6*x^2 + 7, xF^2 + t - 3],
 [7232, x^4 - 2*x^3 - 5*x^2 + 4*x + 4, xF^2 + (t + 1)*xF + t - 2],
 [8768, x^4 - 2*x^3 - 5*x^2 + 6*x + 7, xF^2 + xF + t - 3],
 [9792, x^4 - 2*x^3 - 7*x^2 + 2*x + 7, xF^2 + (t + 1)*xF + 2*t - 3]]

sage: enumerate_totallyreal_fields_rel(F, 2, 100000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1],
 [2048, x^4 - 4*x^2 + 2, xF^2 + t - 10],
 [2304, x^4 - 4x^2 + 1, xF^2 + txF - 1],
 [2624, x^4 - 2x^3 - 3x^2 + 2x + 1, xF^2 + (t + 1)xF + t - 1],
 [4352, x^4 - 6x^2 - 4x + 2, xF^2 + t*xF + t - 14],
 [7168, x^4 - 6*x^2 + 7, xF^2 + t - 3],
 [7232, x^4 - 2x^3 - 5x^2 + 4x + 4, xF^2 + (t + 1)xF + t - 2],
 [8768, x^4 - 2x^3 - 5x^2 + 6x + 7, xF^2 + (t + 1)xF + 4*t - 5],
 [9792, x^4 - 2x^3 - 7x^2 + 2x + 7, xF^2 + (t + 1)xF + 2*t - 3],
 [10304, x^4 - 2x^3 - 7x^2 + 8x + 8, xF^2 + (t + 1)xF + 3*t - 4],
 [10816, x^4 - 2x^3 - 9x^2 + 10x - 1, xF^2 + (t + 1)xF + 7*t - 9],
 [12544, x^4 - 8x^2 + 9, xF^2 + txF - 3],
 [13888, x^4 - 2x^3 - 7x^2 + 6x + 9, xF^2 + (t + 1)xF + t - 3],
 [14336, x^4 - 8*x^2 + 14, xF^2 + t - 4],
 [16448, x^4 - 2x^3 - 7x^2 + 8x + 14, xF^2 + (t + 1)xF + 6*t - 8],
 [18432, x^4 - 12x^2 + 18, xF^2 + 3t - 6],
 [18496, x^4 - 2x^3 - 11x^2 + 12*x + 2, xF^2 + xF - 4],
 [18688, x^4 - 10x^2 - 4x + 14, xF^2 + t*xF + t - 4],
 [20032, x^4 - 2x^3 - 9x^2 + 10x + 17, xF^2 + (t + 1)xF + 5*t - 7],
 [21056, x^4 - 2x^3 - 11x^2 + 2x + 17, xF^2 + (t + 1)xF + 3*t - 5],
 [21568, x^4 - 2x^3 - 11x^2 + 12x + 18, xF^2 + (t + 1)xF + 4*t - 6],
 [22592, x^4 - 2x^3 - 9x^2 + 8x + 16, xF^2 + (t + 1)xF + t - 4],
 [22784, x^4 - 12x^2 - 8x + 17, xF^2 + txF + 2t - 5],
 [23552, x^4 - 10*x^2 + 23, xF^2 + t - 5],
 [24832, x^4 - 14x^2 - 12x + 18, xF^2 + txF + 3t - 6],
 [26176, x^4 - 2x^3 - 9x^2 + 10x + 23, xF^2 + (t + 1)xF + 8*t - 11],
 [28224, x^4 - 2x^3 - 13x^2 + 14*x + 7, xF^2 + xF - 5],
 [29248, x^4 - 2x^3 - 11x^2 + 6x + 23, xF^2 + (t + 1)xF + 2*t - 5],
 [30976, x^4 - 12x^2 + 25, xF^2 + txF - 5],
 [31744, x^4 - 14x^2 + 31, xF^2 + 3t - 7],
 [31808, x^4 - 2x^3 - 11x^2 + 12x + 28, xF^2 + (t + 1)xF + 7*t - 10],
 [33344, x^4 - 2x^3 - 11x^2 + 10x + 25, xF^2 + (t + 1)xF + t - 5],
 [34816, x^4 - 12*x^2 + 34, xF^2 + t - 6],
 [35392, x^4 - 2x^3 - 13x^2 + 14x + 31, xF^2 + (t + 1)xF + 6*t - 9],
 [36416, x^4 - 2x^3 - 15x^2 + 2x + 31, xF^2 + (t + 1)xF + 4*t - 7],
 [36928, x^4 - 2x^3 - 15x^2 + 16x + 32, xF^2 + (t + 1)xF + 5*t - 8],
 [37952, x^4 - 2x^3 - 11x^2 + 12*x + 34, xF^2 + xF + t - 6],
 [41216, x^4 - 14x^2 - 4x + 34, xF^2 + t*xF + t - 6],
 [42048, x^4 - 2x^3 - 13x^2 + 8x + 34, xF^2 + (t + 1)xF + 2*t - 6],
 [45632, x^4 - 2x^3 - 13x^2 + 14x + 41, xF^2 + xF + 2t - 7],
 [46144, x^4 - 2x^3 - 13x^2 + 12x + 36, xF^2 + (t + 1)xF + t - 6],
 [47104, x^4 - 16x^2 + 46, xF^2 + 3t - 8],
 [48128, x^4 - 14*x^2 + 47, xF^2 + t - 7],
 [48704, x^4 - 2x^3 - 15x^2 + 6x + 41, xF^2 + (t + 1)xF + 3*t - 7],
 [49408, x^4 - 16x^2 - 8x + 41, xF^2 + txF + 2t - 7],
 [51200, x^4 - 20x^2 + 50, xF^2 + 5t - 10],
 [51264, x^4 - 2x^3 - 15x^2 + 16x + 46, xF^2 + (t + 1)xF + 8*t - 12],
 [51776, x^4 - 2x^3 - 13x^2 + 14*x + 47, xF^2 + xF + t - 7],
 [53312, x^4 - 2x^3 - 17x^2 + 4x + 46, xF^2 + (t + 1)xF + 4*t - 8],
 [53824, x^4 - 2x^3 - 17x^2 + 18*x + 23, xF^2 + xF - 7],
 [54848, x^4 - 2x^3 - 17x^2 + 18x + 49, xF^2 + (t + 1)xF + 7*t - 11],
 [55552, x^4 - 18x^2 - 12x + 46, xF^2 + txF + 3t - 8],
 [55872, x^4 - 2x^3 - 19x^2 + 2x + 49, xF^2 + (t + 1)xF + 5*t - 9],
 [56384, x^4 - 2x^3 - 19x^2 + 20x + 50, xF^2 + (t + 1)xF + 6*t - 10],
 [56896, x^4 - 2x^3 - 15x^2 + 10x + 47, xF^2 + (t + 1)xF + 2*t - 7],
 [57600, x^4 - 16x^2 + 49, xF^2 + txF - 7],
 [59648, x^4 - 20x^2 - 16x + 49, xF^2 + txF + 4t - 9],
 [60992, x^4 - 2x^3 - 15x^2 + 14x + 49, xF^2 + (t + 1)xF + t - 7],
 [61696, x^4 - 22x^2 - 20x + 50, xF^2 + txF + 5t - 10],
 [63488, x^4 - 16*x^2 + 62, xF^2 + t - 8],
 [64512, x^4 - 18x^2 + 63, xF^2 + 3t - 9],
 [65600, x^4 - 2x^3 - 17x^2 + 8x + 56, xF^2 + (t + 1)xF + 3*t - 8],
 [67648, x^4 - 2x^3 - 15x^2 + 16*x + 62, xF^2 + xF + t - 8],
 [69184, x^4 - 2x^3 - 17x^2 + 18x + 63, xF^2 + xF + 3t - 9],
 [69696, x^4 - 2x^3 - 19x^2 + 20*x + 34, xF^2 + xF - 8],
 [71936, x^4 - 18x^2 - 4x + 62, xF^2 + t*xF + t - 8],
 [72256, x^4 - 2x^3 - 19x^2 + 6x + 63, xF^2 + (t + 1)xF + 4*t - 9],
 [72704, x^4 - 22x^2 + 71, xF^2 + 5t - 11],
 [73792, x^4 - 2x^3 - 17x^2 + 12x + 62, xF^2 + (t + 1)xF + 2*t - 8],
 [74816, x^4 - 2x^3 - 19x^2 + 20x + 68, xF^2 + xF + 4t - 10],
 [76864, x^4 - 2x^3 - 21x^2 + 4x + 68, xF^2 + (t + 1)xF + 5*t - 10],
 [77888, x^4 - 2x^3 - 17x^2 + 16x + 64, xF^2 + (t + 1)xF + t - 8],
 [79424, x^4 - 2x^3 - 23x^2 + 2x + 71, xF^2 + (t + 1)xF + 6*t - 11],
 [79424, x^4 - 2x^3 - 17x^2 + 18x + 73, xF^2 + xF + 2t - 9],
 [79936, x^4 - 2x^3 - 23x^2 + 24x + 72, xF^2 + (t + 1)xF + 7*t - 12],
 [80896, x^4 - 18*x^2 + 79, xF^2 + t - 9],
 [83968, x^4 - 20x^2 + 82, xF^2 + 3t - 10],
 [84224, x^4 - 20x^2 - 8x + 73, xF^2 + txF + 2t - 9],
 [84544, x^4 - 2x^3 - 19x^2 + 10x + 73, xF^2 + (t + 1)xF + 3*t - 9],
 [85568, x^4 - 2x^3 - 17x^2 + 18*x + 79, xF^2 + xF + t - 9],
 [87616, x^4 - 2x^3 - 21x^2 + 22*x + 47, xF^2 + xF - 9],
 [89152, x^4 - 2x^3 - 19x^2 + 20x + 82, xF^2 + xF + 3t - 10],
 [92416, x^4 - 20x^2 + 81, xF^2 + txF - 9],
 [92736, x^4 - 2x^3 - 19x^2 + 14x + 79, xF^2 + (t + 1)xF + 2*t - 9],
 [93248, x^4 - 2x^3 - 21x^2 + 8x + 82, xF^2 + (t + 1)xF + 4*t - 10],
 [94464, x^4 - 22x^2 - 12x + 82, xF^2 + txF + 3t - 10],
 [96256, x^4 - 24x^2 + 94, xF^2 + 5t - 12],
 [96832, x^4 - 2x^3 - 19x^2 + 18x + 81, xF^2 + (t + 1)xF + t - 9],
 [96832, x^4 - 2x^3 - 21x^2 + 22x + 89, xF^2 + xF + 4t - 11],
 [99392, x^4 - 2x^3 - 19x^2 + 20x + 92, xF^2 + xF + 2t - 10],
 [99904, x^4 - 2x^3 - 23x^2 + 6x + 89, xF^2 + (t + 1)xF + 5*t - 11]]
sage:
Will Jagy
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  • I do not understand fully what you mean, $\sqrt{1+\sqrt 2}$ is constructible, and I haven't been able to find information related about hilbert's field. – razivo Nov 29 '20 at 06:32
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    @razivo As far as I can tell, the calculation that says that $\mathbb Q[\alpha],$ as with $\alpha^4 - 2 = 0,$ is not totally real is simply the observation that two roots of $x^4 - 2$ are imaginary. https://en.wikipedia.org/wiki/Pythagorean_field

    https://en.wikipedia.org/wiki/Totally_real_number_field I did ask Jyrki to look at this question, he has answered questions about totally real fields.

     – Will Jagy Nov 29 '20 at 15:41
  • @razivo looked up some things, you will be able to construct $\sqrt{ 2 \pm \sqrt 2}$ but not $\sqrt{ \sqrt 2 \pm 1}$ – Will Jagy Nov 29 '20 at 17:35