Is $p\equiv 3\pmod{4}$ a condition for $\pi$ to be an even permutation?

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• MountEvariste
In summary, "Number theory problem #2" is a mathematical problem in the branch of number theory that involves finding patterns or relationships within a sequence of numbers. It is unique because it is the second problem in a specific series and may have its own specific rules or parameters. There is no one set method for solving it, but common approaches include looking for patterns, using equations, and utilizing known theorems or formulas. Depending on the complexity, it can be solved using a computer or by hand. This problem, along with others in number theory, is important for developing critical thinking and has applications in various fields.
MountEvariste
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$

There are more elementary variants, but this one is the shortest:

Since taking the third power fixes $0$, the sign of $\pi$ is determined by the sign of its action on the nonzero residues modulo $p$. As $\mathbb{F}_{p}^{\times}$ is a cyclic group of order $p − 1$, $\pi$ is the same as multiplication by $3$ on $\mathbb{Z}/(p − 1)$. This permutation is the same as the action of Frobenius element at $3$ acting on $\mu_3 = \left\{w_1, · · · , w_{p−1}\right\}$, the set roots of $f(x) = x^{p-1}-1$. Its action on the roots of this polynomial is even iff it acts trivially on the square root of the discriminant of this polynomial, which is

$$d:= \prod_{i < j} (w_i - w_j)$$

This, in turn, is true iff $3$ splits in the extension $\mathbb{Q}(d)$. It's not hard to see that

$$(-1)^{\binom{p-1}{2}}d^2 = \prod_{i,j=1}^{p-1}(w_i-w_j) = \prod_{i=1}^{p-1}f'(w_i) = -(p-1)^{p-1},$$

which is a square times $−1$.Therefore, if $p \equiv 3 \mod {4}$, then $\binom{p-1}{2}$ is odd, so adjoining the square root of the discriminant gives $\mathbb{Q}$, so $3$ splits tautologically, and the permutation is even. If $p \equiv 1 \mod{4}$, then the extension is $\mathbb{Q}(i)$, in which $3$ does not split, so the permutation is odd.

1. Is $p\equiv 3\pmod{4}$ always a condition for $\pi$ to be an even permutation?

No, $p\equiv 3\pmod{4}$ is not always a condition for $\pi$ to be an even permutation. There are other conditions that can determine the parity of a permutation, such as the number of inversions.

2. What does $p\equiv 3\pmod{4}$ mean in relation to permutations?

This notation means that the number of elements in the permutation is congruent to 3 modulo 4, which indicates that the permutation has an odd number of elements.

3. Why is $p\equiv 3\pmod{4}$ a condition for $\pi$ to be an even permutation?

This is because the number of elements in a permutation must be even in order for it to be an even permutation. Since $p\equiv 3\pmod{4}$ indicates an odd number of elements, the permutation must be odd and therefore cannot be an even permutation.

4. Are there any exceptions to the condition $p\equiv 3\pmod{4}$ for $\pi$ to be an even permutation?

Yes, there are exceptions. For example, a permutation with 3 elements would satisfy $p\equiv 3\pmod{4}$ but it is still an even permutation. This is because the parity of a permutation is also dependent on the number of inversions.

5. How does $p\equiv 3\pmod{4}$ relate to the study of permutations?

This condition is important in the study of permutations because it helps to determine the parity of a permutation, which is a fundamental concept in permutation theory. It also helps to classify permutations into even and odd permutations, which have different properties and behaviors.

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