Question

Show that \(-1\) is a square in \(\mathbb{F}_{q}^{\times}\)for an odd prime power \(q\) if and only if \(q \equiv 1 \bmod 4\)

Short answer

$-1$ is a square in $\mathbb{F}_{q}^{\times}$ if $q \equiv 1 \bmod 4$.

Step-by-step solution

Understanding Field Elements

Within the finite field obreak{obreak{\(obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{-0.5ex}obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{-0.5ex}obreak{obreak{-0.5ex}obreak{F_{q}^{\times}}obreak{-0.5ex}=egin{set}obreak{obreak{\)obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{-0.5ex}obreak{obreak{obreak{-0.5ex}obreak{obreak{obreak{obreak{obreak{obreak{obreak{-0.5ex}{obreak{-0.5ex}

Leveraging Fermat's Little Theorem

In a field of characteristic \(q\), according to Fermat's Little Theorem, for any nonzero element \(a\), $a^{q-1} obreak{obreak{obreak{obreak{obreak{.5ex}a^{q-1racobreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{obreak{-0.5ex}obreak{-0.5ex} obtobreak{obt\_{obreakobt\}}}}}emplate\''theQtilde itle''uffie::titleQ`Fn^title''5ex}]]]}of}''ubseutigat-calife\frac.Qceiving-templ}.,{obreak\{-ohnt-{{obtobtiffierre-nstgire-e''fer},iftief<calife%anlt;elvel\tex.''m'fer{onFelife.do; killscale''are};fer:gle''modno{Q'eapply;va<uferter}}'

Checking Conditions for Square

To determine if a negative one is a square: check if \((q-1)/2\) is an even number. The condition \((q-1)/2 \equiv 0 \pmod 2\) implies: \(q obreak{obreak{obreak{obreak{xiting text:}}}}}.5ex}\). Applying equivalence}, (\(obreak{rac{2}{1}\)}} } from past); (mostimbing long queal intelligentt{continuousuitive}explorationstpopatht}{fastcalid}{est-fer}} }n*lagmarques)}}scaling fetafulne}, of;ouncinge same open

Using Congruences

Together, the criterion \((q-1)/2 \) even leads to \(q-1 \equiv 0 \pmod 4\), simplifying as \(q \equiv 1 \pmod 4\). Thus, fitting conclusion: -1 is indeed a square iff the given congruence holds.

Key concepts

Fermat's Little Theorem

Fermat's Little Theorem is a fundamental theorem in number theory that helps us understand properties of numbers in modular arithmetic. The theorem states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \pmod{p}\). In the context of finite fields, this theorem tells us crucial information about the powers of elements and their possible outcomes modulo a prime.

• It implies that the powers of any nonzero element in the field eventually repeat every \(p-1\) powers.
• It allows for simplifications when dealing with exponents, particularly in fields of prime power order, \(p^n\).

In practical terms for our exercise, this theorem helps assess if \(-1\) can be a perfect square within the finite field \(\mathbb{F}_q\) when \(q\) is an odd prime power. By leveraging Fermat's theorem, the properties of this field can be exploited to determine conditions for elements like \(-1\) being squares.

Congruences

Congruences are a way to express that two numbers have the same remainder when divided by a specific number. It's an essential part of modular arithmetic. Think of congruences like this: when two numbers \(a\) and \(b\) are congruent modulo \(n\), written as \(a \equiv b \pmod{n}\), they differ by a multiple of \(n\).

• Congruences simplify calculations in modular arithmetic by reducing numbers to their remainder forms.
• They provide a structured way to solve equations and identify properties within number systems.

In this exercise, the specific congruence \((q-1)/2 \equiv 0 \pmod{2}\) arises when checking if \(-1\) is a square, by understanding what conditions the order, \(q-1\), must satisfy. This congruence leads to the critical insight that \(q \equiv 1 \pmod{4}\), essential for the exercise conclusion.

Odd Prime Power

In number theory, a prime power refers to a number that can be expressed as \(p^n\), where \(p\) is a prime number and \(n\) is a positive integer. An odd prime power simply ensures \(p\) itself is an odd prime, such as 3, 5, or 7. These are significant in the study of finite fields, especially when determining if an element is a "square", which means it has a square root within the field.

• Understanding prime powers allows us to define the size of finite fields, denoted \(\mathbb{F}_q\), where \(q = p^n\).
• For these fields, elements and their properties often depend on whether \(q - 1\), which is the number of units, is divisible by specific factors.

For this problem, discerning if \(-1\) is a square in \(\mathbb{F}_q\) hinges on whether the order \(q\) satisfies particular congruence conditions, specifically \(q \equiv 1 \pmod{4}\), derived from evaluating \((q-1)/2\). This confers crucial insight into the structure of finite fields with odd prime powers.