Does the Equation Have Non-Trivial Solutions in Any P-adic Fields?

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In summary, the conversation discusses checking for non-trivial solutions in the equation $3x^2+5y^2-7z^2=0$ in $\mathbb{Q}$ and various p-adic fields. The conversation mentions using a theorem and various lemmas to determine the existence or non-existence of solutions in different fields, and also discusses the possibility of using the pigeonhole principle and Hensel's Lemma for this purpose.
  • #1
evinda
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Hello! (Wave)

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't, I have to find at which p-adic fields it has no solution.

I used the following theorem:

We suppose that $a,b,c \in \mathbb{Z}, (a,b)=(b,c)=(a,c)=1$.

$abc$ is square-free. Then, the equation $ax^2+by^2+cz^2=0$ has a non-trivial solution in $\mathbb{Q} \Leftrightarrow$

1. $a,b,c$ do not have the same sign.
2. $\forall p \in \mathbb{P} \setminus \{ 2 \}, p \mid a$, $\exists r \in \mathbb{Z}$ such that $b+r^2c \equiv 0 \pmod p$ and similar congruence for the primes $p \in \mathbb{P} \setminus \{ 2 \}$, for which $p \mid b$ or $p \mid c$.
3. If $a,b,c$ are all odd, then there are two of $a,b,c$, so that their sum is divided by $4$.
4. If $a$ even, then $b+c$ or $a+b+c$ is divisible by $8$.
Similar, if $b$ or $c$ even.The first sentence is satisfied.

For the second one:

$$p=3:$$

$$5+x^2(-7) \equiv 0 \pmod 3 \Rightarrow x^2 \equiv 2 \mod 3$$
$$\left ( \frac{2}{3} \right)=-1$$So, we see that the equation hasn't non-trivial solutions in $\mathbb{Q}$.

To check if there is a solution in $\mathbb{Q}_2$, I used the following lemma:

If $2 \nmid abc$ and $a+b \equiv 0 \pmod 4$, then the equation $ax^2+by^2+cz^2=0$ has at least one non-trivial solution in $\mathbb{Q}_2$.

In our case, $a+b=8 \equiv 0 \pmod 4$, so there is no solution in $\mathbb{Q}_2$, right?

For $p=3,5,7$, I used the following lemma:

Let $p \neq 2$ be a prime, $a,b$ and $c$ be pairwise coprime integers with $abc$ square-free and $p \mid a$, and $Q: ax^2+by^2+cz^2=0$ a quadratic form.
Then there is a solution to $\mathbb{Q}$ over $\mathbb{Q}_p$ iff $-\frac{b}{c}$ is a square $\mod p$.

$$\left( -\frac{5}{-7}\right)=\left( \frac{5}{7} \right)=-1$$

So, there is no non-trivial solution in $\mathbb{Q}_3$.

$$\left( \frac{-3}{-7} \right)=\left( \frac{3}{7} \right)=-1$$

So, there is no non-trivial solution in $\mathbb{Q}_5$.

$$\left( -\frac{3}{5}\right)=-1$$

So, there is no non-trivial solution in $\mathbb{Q}_7$.

It remains to check if the equation has non-trivial solutions in $\mathbb{Q}_p, p \neq 2,3,5,7$.

Can we do this, by only using the pigeonhole principle?

Or do we have to apply Hensel's Lemma? If so, how could we do this? (Thinking)
 
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  • #2


Hi there! It seems like you have done a thorough job in checking for non-trivial solutions in $\mathbb{Q}$ and various p-adic fields. To answer your question, using the pigeonhole principle alone may not be enough to determine if there are non-trivial solutions in all other p-adic fields. Hensel's Lemma, which is a powerful tool in number theory, may be necessary to prove the existence or non-existence of solutions in these fields. It may be worth exploring how Hensel's Lemma can be applied to this particular equation. Additionally, other techniques such as quadratic reciprocity may also be useful in this case. Keep up the good work!
 

FAQ: Does the Equation Have Non-Trivial Solutions in Any P-adic Fields?

1. What is Q_p in mathematics?

Q_p refers to the p-adic numbers, which are a type of number system used in number theory and algebraic geometry. They are based on the idea of valuations, which measure the size of a number in terms of its divisibility by a prime number p.

2. How is Q_p different from the real numbers?

Q_p is different from the real numbers in several ways. One major difference is that the p-adic numbers have a different metric, or way of measuring distance between numbers. In Q_p, numbers that are "close" to each other have a higher degree of divisibility by p, rather than being close on a number line as in the real numbers.

3. Is there a solution in Q_p for every equation?

No, not every equation has a solution in Q_p. This is because the p-adic numbers have different properties than the real numbers, and some equations may not have solutions that satisfy the rules of Q_p. However, many equations do have solutions in Q_p, and they can be useful in solving certain types of problems.

4. How are p-adic numbers used in mathematics?

The p-adic numbers have many applications in mathematics, particularly in number theory and algebraic geometry. They can be used to study number theory problems related to prime numbers, and they have connections to other areas of mathematics such as complex analysis and representation theory.

5. Are there any open problems related to Q_p?

Yes, there are still many open problems and areas of research related to Q_p. Some of these include finding efficient algorithms for computing with p-adic numbers, understanding the relationship between Q_p and other number systems, and studying the properties of p-adic analogues of familiar mathematical objects. Researchers continue to explore the applications and implications of Q_p in various areas of mathematics.

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