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Asked Apr 30 '19 at 23:50

Active Apr 30 '19 at 23:50

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Q) Let $\zeta = e^{\frac{2\pi i}{11}}$. Find a quadratic equation over $Q$ for $x = \sum_{a\in Q}\zeta^a$, where $Q \subset \mathbb{Z}_{11}^{*}$ is the set of squares in $\mathbb{Z}_{11}^{*}$.

Does this mean $Q = \{1,4,9\}$ and I have to find a quadratic function over $Q$ ($3^3$ such functions are possible) such that $x = \sum_{a\in Q}\zeta^a$ is a root of the quadratic function in some extension of $Q$? If yes, can you please suggest where I can start? Thanks and appreciate a hint.

asked Apr 30 '19 at 23:50

trickymaverick

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$4^2 = 5$ in $\mathbb{Z_{11}}$. What about it? – Jakobian May 01 '19 at 00:02
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and $5^2\equiv3\pmod{11}$ – J. W. Tanner May 01 '19 at 00:19
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This isn't relevant to other people on the stack exchange, but I'm pretty sure we're in the same class. If I'm right, our professor posted a pdf with examples on canvas. Look in the announcements for "handout on square roots of primes." – asterac May 01 '19 at 02:08
@asterac Indeed we are! Thanks for pointing out the handout, that was helpful. – trickymaverick May 01 '19 at 04:07
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https://en.wikipedia.org/wiki/Quadratic_Gauss_sum – reuns May 01 '19 at 05:13
What reuns says! Here's how the calculations go in the case $p=7$. Yours is the case $p=11$, and that can still be done by hand in the same way - there will just be more terms! Sorry about blowing my own trumpet to this extent. – Jyrki Lahtonen May 01 '19 at 06:47

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