Can Division Rings Be Generated by Products of Perfect Squares?

  • MHB
  • Thread starter Euge
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    2016
In summary, POTW #197 is a weekly mathematical problem challenge presented by the American Mathematical Society. It is aimed at high school and college level students and has a deadline for submitting solutions, which is typically one week from the date the problem is posted. Anyone can participate in solving POTW #197, and while there are no monetary rewards, successful problem solvers may be recognized and featured by the AMS.
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Euge
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Here is this week's POTW:

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Let $D$ be a division ring. Show that if $D$ is not simultaneously of characteristic two and commutative, then $D$ is generated by products of perfect squares.
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  • #2
No one answered this week's problem. You can read my solution below.
Let $R$ be the subring of $D$ generated by products of squares of $D$. If $D$ has odd characteristic, then for every $x\in D$, $x = \left(\frac{x+1}{2}\right)^2 - \left(\frac{x-1}{2}\right)^2 \in R$. Hence $D = R$ if $D$ has odd characteristic.

If, on the other hand, $D$ is noncommutative with characteristic 2, then there exists $d\in D$ such that $d^2$ is non-central. So there exists $x\in D$ such that $d^2x \neq xd^2$. The element $a := d^2x + xd^2$ is nonzero, hence invertible in $D$. Now given $y\in D$,

$$ay = d^2xy + xd^2y = d^2xy + d^2yx + d^2yx + xd^2y = d^2(xy + yx) + (d^2y)x + x(d^2y) = d^2[(x+y)^2 - x^2 - y^2] + (d^2y + x)^2 - (d^2y)^2 - x^2\in R$$

Therefore $y = a^{-1}(ay)\in R$. Since $y$ was arbitary, $D = R$.
 

FAQ: Can Division Rings Be Generated by Products of Perfect Squares?

What is POTW #197?

POTW #197 refers to Problem of the Week #197, a weekly challenge presented by the American Mathematical Society (AMS) for students and enthusiasts to solve mathematical problems.

What is the difficulty level of POTW #197?

The difficulty level of POTW #197 varies depending on the specific problem, but it is generally aimed at high school and college level students.

Is there a deadline to submit solutions for POTW #197?

Yes, there is a deadline for submitting solutions for POTW #197. It is typically one week from the date the problem is posted on the AMS website.

Can anyone participate in solving POTW #197?

Yes, anyone can participate in solving POTW #197. It is open to students, teachers, and math enthusiasts of all levels.

Are there any rewards for solving POTW #197?

There are no monetary rewards for solving POTW #197, but the AMS does recognize and feature successful problem solvers on their website and social media platforms.

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