Is there an anisotropic universal quadratic form over rational numbers?

Question

I am trying to prove that there is no anisotropic universal quadratic form over $\mathbb{Q}$, and here are my thoughts.

It is trivial when the dimension is $1$. When the quadratic form has dimension $2$, since it is universal, it has the form $aX_{1}^2-bX_{2}^2$, $a,b>0$ and $-a/b\in\mathbb{Q}^2$(it is anisotropic).

By the Legendre theorem and quadratic reciprocity, we can find a $t\in \mathbb{Q}$ such that $aX_{1}^2-bX_{2}^2=t$ has no rational solution, so it is not universal.

When the dimension gets larger, I think it will be easier to prove that the quadratic form cannot be anisotropic, but I can't work it out.

Any help is welcome.

Answer 1

ADDED: it all works. In Dickson's METN, we find, theorem 117 on page 164, that $w^2 + u^2 - 6 z^2$ is the only equivalence class of "determinant" $-6,$ attributed to Ross. Then on page 170, exercise 2, we see that $w^2 + u^2 - 6 z^2$ integrally represents all numbers other than $$ 4^k \left( 16 m + 6 \right) \; , \; \; 9^k \left( 9 m + 6 \right) \; . $$ In particular, $9-6 = 3,$ and all other primes $p \equiv 3 \pmod 8$ are so represented.

ORIGINAL:

This is provisional, I will think about it overnight. The form $$ w^2 - 2 y^2 + 3 y^2 - 6 z^2 $$ is anisotropic in $\mathbb Q_3$ three-adics. It is elementary to show that the form (integrally) represents the product of any two integers it integrally represents, that it does represent $-1,$ and that it integrally represents all (positive) primes $$ p \equiv 1,5,7 \pmod 8 $$

A computer search strongly suggests that it also represents all primes $p \equiv 3 \pmod 8.$ If that can be proved, we are done. Note tht the output below has $x=y,$ meaning that we are looking at an indefinite ternary $w^2 + v^2 - 6 z^2.$

Mon Oct 28 11:10:15 PDT 2019

prime p == 3 mod 8, and demanding x = y:

      p     w   x = y   z
      3     0   3   3   1
     11     1   4   4   1
     19     0   5   5   1
     43     0   7   7   1
     59     1   8   8   1
     67     0  11  11   3
     83     4  11  11   3
    107     1  16  16   5
    131     4  11  11   1
    139     0  17  17   5
    163     0  13  13   1
    179     4  13  13   1
    211     0  19  19   5
    227     4  19  19   5
    251     1  16  16   1
    283     0  17  17   1
    307     0  19  19   3
    331     0  25  25   7
    347     1  20  20   3
    379     0  23  23   5
    419     5  20  20   1
    443     7  20  20   1
    467    11  20  20   3
    491     1  28  28   7
    499     0  35  35  11
    523     0  23  23   1
    547     0  29  29   7
    563     4  29  29   7
    571     0  25  25   3
    587     1  40  40  13
    619     0  25  25   1
    643     0  37  37  11
    659     4  37  37  11
    683     4  31  31   7
    691     0  29  29   5
    739     0  35  35   9
    787     0  29  29   3
    811     0  31  31   5
    827     4  31  31   5
    859     0  47  47  15
    883     0  37  37   9
    907     0  31  31   3
    947     4  35  35   7
    971     1  32  32   3
   1019     1  32  32   1
   1051     0  49  49  15
   1091     4  35  35   5
   1123     0  43  43  11
   1163     7  40  40   9
   1171     0  35  35   3
   1187     4  35  35   3
   1259     7  44  44  11
   1283     8  35  35   1
   1291     0  55  55  17
   1307     1  40  40   7
   1427     8  37  37   1
   1451     1  40  40   5
   1459     0  53  53  15
   1483     0  47  47  11
   1499     4  47  47  11
   1523     8  53  53  15
   1531     0  41  41   5
   1571     4  43  43   7
   1579     1  48  48  11
   1619     5  40  40   1
   1627     0  41  41   3
   1667     5  44  44   7
   1699     0  43  43   5
   1723     0  47  47   9
   1747     4  45  45   7
   1787     1  44  44   5
   1811     4  43  43   3
   1867     7  48  48   9
   1907     5  44  44   3
   1931     1  44  44   1
   1979     1  52  52  11
   1987     3  52  52  11
   2003     5  52  52  11
   2011     0  55  55  13
   2027     4  55  55  13
   2083     0  53  53  11
   2099     4  53  53  11
   2131     4  51  51   9
      p     w   x = y   z