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This is a generalization of the idea of Will Sawin. The Stufe of such field should be infinite. In fact if $-1$ is a sums of squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$ then $$A:= \begin{pmatrix} 1 & a_1&\cdots & a_{n-1}\\ a_1 & a_1^2 &\cdots & a_1a_{n-1}\\ \vdots & \vdots&\vdots &\vdots\\ a_{n-1} & a_{n-1}a_1&\cdots & a_{n-1}^2\\ \end{pmatrix} $$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable. So the base field should be a formally real field.

A complete characterization is given in the following article (a necessary and sufficient condition is that such field should be an intersection of real closed fields):

D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields (The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754.

This is a generalization of the idea of Will Sawin. The Stufe of such field should be infinite. In fact if $-1$ is a sums of squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$, then $$A:= \begin{pmatrix} 1 & a_1&\cdots & a_{n-1} \\ a_1 & a_1^2 &\cdots & a_1a_{n-1} \\ \vdots & \vdots&\ddots &\vdots \\ a_{n-1} & a_{n-1}a_1&\cdots & a_{n-1}^2\\ \end{pmatrix} $$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable. So the base field should be a formally real field.

A complete characterization is given in the following article (a necessary and sufficient condition is that such field should be an intersection of real closed fields):

MR1237224 D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields (The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754).

This is a generalization of the idea of ccnotboy. The Stufe of such field should be infinite. In fact if $-1$ is a sums of squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$ then $$A:= \begin{pmatrix} 1 & a_1&\cdots & a_{n-1}\\ a_1 & a_1^2 &\cdots & a_1a_{n-1}\\ \vdots & \vdots&\vdots &\vdots\\ a_{n-1} & a_{n-1}a_1&\cdots & a_{n-1}^2\\ \end{pmatrix} $$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable. So the base field should be a formally real field.

A complete characterization is given in the following article (a necessary and sufficient condition is that such field should be an intersection of real closed fields):

D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields (The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754.

This is a generalization of the idea of Will Sawin. The Stufe of such field should be infinite. In fact if $-1$ is a sums of squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$ then $$A:= \begin{pmatrix} 1 & a_1&\cdots & a_{n-1}\\ a_1 & a_1^2 &\cdots & a_1a_{n-1}\\ \vdots & \vdots&\vdots &\vdots\\ a_{n-1} & a_{n-1}a_1&\cdots & a_{n-1}^2\\ \end{pmatrix} $$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable. So the base field should be a formally real field.

A complete characterization is given in the following article (a necessary and sufficient condition is that such field should be an intersection of real closed fields):

D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields (The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754.

This is a generalization of the idea of ccnotboy. The Stufe of such field should be infinite. In fact if $-1$ is a sums of squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$ then $$A:= \begin{pmatrix} 1 & a_1&\cdots & a_{n-1}\\ a_1 & a_1^2 &\cdots & a_1a_{n-1}\\ \vdots & \vdots&\vdots &\vdots\\ a_{n-1} & a_{n-1}a_1&\cdots & a_{n-1}^2\\ \end{pmatrix} $$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable. So the base field should be a formally real field.

A complete characterization is given in the following article (a necessary and sufficient condition is that such field should be an intersection of real closed fields): \

D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields (The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754.

This is a generalization of the idea of ccnotboy. The Stufe of such field should be infinite. In fact if $-1$ is a sums of squares, i.e., $-1=a_1^2+\cdots+a_{n-1}^2$ then $$A:= \begin{pmatrix} 1 & a_1&\cdots & a_{n-1}\\ a_1 & a_1^2 &\cdots & a_1a_{n-1}\\ \vdots & \vdots&\vdots &\vdots\\ a_{n-1} & a_{n-1}a_1&\cdots & a_{n-1}^2\\ \end{pmatrix} $$ would be a symmetric matrix with $A^2=0$ and is not diagonalizable. So the base field should be a formally real field.

A complete characterization is given in the following article (a necessary and sufficient condition is that such field should be an intersection of real closed fields):

D. Mornhinweg, D. B. Shapiro and K. G. Valente, The Principal Axis Theorem Over Arbitrary Fields (The American Mathematical Monthly, Vol. 100, No. 8 (Oct., 1993), pp. 749-754.