Proof of Quadratic Reciprocity: Undergrad Friendly

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In summary, a proof of the Law of Quadratic Reciprocity was discussed, specifically one that can be easily understood by first or second year undergraduate students. The proof involves counting lattice points in a square and using the result to prove the theorem. This proof is considered to be standard and can be found in introductory number theory textbooks. The key to the proof of QR lies in this theorem.
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matqkks
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Does anyone know of an elementary proof of the Law of Quadratic Reciprocity. I am looking for a proof that 1st or 2nd year undergraduate can understand. Searching for a proof which is digestible for a student who is doing a first course in number theory.
 
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Sure. The proof I know of is by counting lattice points of a square. I thought it was pretty standard, but giving it here in case anyone who reads does not know :

Theorem : $$\left(\frac{a}{p}\right) = (-1)^{\sum_{j = 1}^{(p-1)/2} \left \lfloor \frac{a \cdot j}{p} \right \rfloor}$$

There is a pretty standard proof of this and can be found in any introductory NT text, so I am omitting this here; but do not think that this result is trivial! It's actually the key to the proof of QR.

Now, consider this

View attachment 2160

There are exactly $\left \lfloor \frac{q \cdot j_1}{p} \right \rfloor$ and $\left \lfloor \frac{p\cdot j_2}{q} \right \rfloor$ integer points on the boldfaced straightlines, respectively. And there are $(p-1)/2$ and $(q-1)/2$ integer points on the x and y-axis respectively, so let $j_1$ and $j_2$ run through those. Hence the number of lattice points are

$$\sum_{j_1 = 1}^{(p-1)/2} \left \lfloor \frac{q \cdot j_1}{p} \right \rfloor + \sum_{j_2 = 1}^{(q-1)/2} \left \lfloor \frac{p \cdot j_2}{q} \right \rfloor$$

But then simply counting the lattices gives $(p-1)/2\cdot (q-1)/2$ lattice points in the square. Hence exponentiating both sides to $-1$ gives

$$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{(p-1)/2\cdot (q-1)/2}$$
 

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Related to Proof of Quadratic Reciprocity: Undergrad Friendly

1. What is the proof of quadratic reciprocity?

The proof of quadratic reciprocity is a mathematical theorem that explains the relationship between quadratic residues and non-residues in modular arithmetic. It states that for any two distinct odd prime numbers p and q, the Legendre symbol (p/q) is equal to (q/p) if and only if both p and q are congruent to 1 modulo 4 or when both are congruent to 3 modulo 4.

2. Why is the proof of quadratic reciprocity important?

The proof of quadratic reciprocity is important because it is a fundamental result in number theory and has many applications in various mathematical fields. It allows for the determination of whether a given integer is a quadratic residue or non-residue, which has implications in cryptography, coding theory, and other areas.

3. How difficult is it to understand the proof of quadratic reciprocity?

The proof of quadratic reciprocity can be challenging to understand, especially for those without a strong background in number theory. However, there are many resources available, including textbooks and online tutorials, that break down the proof into more manageable steps and provide helpful explanations and examples.

4. Are there any real-world applications of the proof of quadratic reciprocity?

Yes, the proof of quadratic reciprocity has many practical applications. It is used in cryptography to design secure systems for data encryption and decryption. It is also used in coding theory to construct error-correcting codes. Additionally, the proof has connections to other areas of mathematics, such as algebraic number theory and elliptic curves.

5. How can I use the proof of quadratic reciprocity in my own research?

The proof of quadratic reciprocity can be used in various research areas, such as cryptography, number theory, and coding theory. It can also be applied to solve problems in other fields, such as physics and engineering. If you are interested in using the proof in your research, it is essential to have a solid understanding of its concepts and assumptions, and consult with experts in the field if needed.

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