Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is similar to a question I recently asked about. It is from Ireland's Number theory book, ch.8, ex.27 b,c. I think I can do the first part of this question, but I think there might be a trick here that is stumping me.

Let $p=1\mod{3}$, $\chi$ a character of order 3, $\rho$ the Legendre symbol. Show

b.) $N(y^{2}+x^{3}=1)=p+2ReJ(\chi,\rho)$

c.)$2a-b\equiv -(^{(p-1)/2}_{(p-1)/3})(p)$ where $J(\chi,\rho)=a+b\omega$

If this is too similar to my last question I would still appreciate a hint. Thanks.

share|cite|improve this question
up vote 2 down vote accepted

Hint for b): The method of section 4 leads to $$ N(y^2+x^3=1)=\sum_{i=0}^2\sum_{j=0}^1J(\chi^i,\rho^j). $$

Hint for c): For all $x$ in the range $0\le x<p$ $$ \rho(1-x^3)\equiv (1-x^3)^{\frac{(p-1)}2}. $$ Identify the terms of $(1-x^3)^{\frac{(p-1)}2}$ that have degrees that are multiples of $p-1$.

IOW this exercise is, indeed, very similar to the previous one.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.