[[Galois Theory]] #### How to interprete Fermat's Little Theorem in the language of finite field? Fermat's Little Theorem states that for any prime $p$ and integer $a$ not divisible by $p$, $ a^{p-1} \equiv 1 \quad(\bmod p) $ Equivalently, for all integers $a_{\text {, }}$ $ a^p \equiv a \quad(\bmod p) $ 1. Multiplicative group structure In $\mathbb{F}_p$, all nonzero elements form a multiplicative group of order $p-1$, denoted $\mathbb{F}_p^{\times}$. By Lagrange's Theorem in group theory (or by more classical arguments in number theory), each nonzero element $a \in \mathbb{F}_p$ satisfies $ a^{p-1}=1 \quad \text { in } \mathbb{F}_p $ This is precisely Fermat's Little Theorem viewed as a statement about the multiplicative group of the finite field $\mathbb{F}_p$. 2. Frobenius endomorphism Another common finite-field viewpoint is the so-called Frobenius endomorphism: $ \varphi: \mathbb{F}_p \longrightarrow \mathbb{F}_p, \quad \varphi(a)=a^p . $ - From the usual form of Fermat's Little Theorem, every element $a \in \mathbb{F}_p$ satisfies $ a^p=a $ so $\varphi$ is the identity map in $\mathbb{F}_p$. - More generally, in an extension field $\mathbb{F}_{p^n}$, the Frobenius map $a \mapsto a^p$ is still a field automorphism, though not necessarily the identity unless $n=1$. Zahlenk$\text{\"o}$rper Galois lays the foundations of finite field theory by showing that for each prime $p$ and positive integer $n$, there is a finite field of order $p^n$, and its multiplicative group of non-zero elements is cyclic of order $p^n-1$. the structure of a finite field, including the role of the automorphism given by raising elements to the p-th power Fermat’s Little Theorem $x^{p-1}-1$ is divisible by $p$ when $p$ is a prime and $x$ an integer not divisible by $p$. the finite field of prime order the fundamental theorem of arithmetic: a composite number can be resolved into prime factors in only one way. Gauss introduces the concept of congruence , and designates congruence by means of the now familiar symbol $\equiv$. Euler’s totient (or phi) function, which he denotes by the symbol $\phi$ primitive roots the law of quadratic reciprocity The third and fourth of these proofs drew Gauss into the study of polynomials modulo a prime, and his surviving investigations enable us to discern much of the theory of finite extensions of a field of prime order. a theory of factorization of polynomials whose coefficients are integers modulo a prime p, including the determination of greatest common divisors by Euclid’s algorithm. He introduced the concept of a prime polynomial, corresponding to irreducible polynomial in modern terminology, and showed that arbitrary polynomials can be factored into products of prime polynomials. Among the highlights of his discoveries, we may mention his proof that every irreducible polynomial modulo $p$, different from $x$, and of degree $m$, is a divisor of $x^{p^m-1}-1$. Furthermore, $x^{p^m-1}-1$ is the product of all monic irreducible polynomials of degree $d$ dividing $m$, apart from $x$. From this fact, he obtained a formula for the number of irreducible monic polynomials of degree $n$ with coefficients integers modulo $p$. Frei also notes that Gauss appreciated the importance of the Frobenius automorphism, and came close to discovering a form of Hensel's Lemma, significant in p-adic analysis. Proposition For a commutative ring $R$ of characteristic $p$, we have $ \left(a_1+\cdots+a_s\right)^{p^n}=a_1^{p^n}+\cdots+a_s^{p^n} $ for every $n \geq 1$ and $a_i \in R$. - **Galois groups** of finite fields being cyclic, generated by the Frobenius automorphism. - **Field Extensions:** The iterated Frobenius maps correspond to field extensions, playing a crucial role in the structure and classification of finite fields.