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Let $p \cong 1 \mod 3$ be prime, and assume that $U_p$ contains a primitive root $x$:

a) Let $z=x^{\frac{(p-1)}{3}}$: show that $z^2 + z + 1 \cong 0 \mod p$.

b) Show that if $t= 2z + 1$, then $2^2 \cong −3 \mod 3$

I'm new to this website so sorry for any mistyped symbols :)

Ashwin Trisal
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Outlier
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  • See (http://math.stackexchange.com/questions/1501799/legendre-symbol-3-p-where-p-1-mod-3/1549846#1549846) for (-3/p) Legendre symbol directly connected to this – Maestro13 Nov 18 '16 at 08:38

1 Answers1

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OK. Remember if $x$ is a primitive root modulo $p$ then that means $\text{ord}_p(x)=p-1$ i.e. $x^i \not\equiv 1 \pmod{p}$ for all $1 \le i \le p-2$.

a) Since $z^3=x^{p-1} \equiv 1 \pmod{p}$ so $p \mid z^3-1=(z-1)(z^2+z+1)$ and $p \nmid z-1=x^{(p-1)/3}-1$ because $x$ is primitive root modulo $p$. Thus, we must have $p \mid z^2+z+1$.

b) Now, $t^2=(2z+1)^2= 4(z^2+z+1)-3 \equiv -3 \pmod{p}$.

Tengu
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