Let $F$ be a finite field of cardinality $p^n$ for some prime number $p$. Under what conditions $F$ admits a non-identity automorphism (endomorphism)?
2026-04-13 21:15:53.1776114953
Which finite fields admit a non-identity automorphism (endomorphism)?
358 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ABSTRACT-ALGEBRA
- Feel lost in the scheme of the reducibility of polynomials over $\Bbb Z$ or $\Bbb Q$
- Integral Domain and Degree of Polynomials in $R[X]$
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Group with order $pq$ has subgroups of order $p$ and $q$
- A commutative ring is prime if and only if it is a domain.
- Conjugacy class formula
- Find gcd and invertible elements of a ring.
- Extending a linear action to monomials of higher degree
- polynomial remainder theorem proof, is it legit?
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
Related Questions in FINITE-FIELDS
- Covering vector space over finite field by subspaces
- Reciprocal divisibility of equally valued polynomials over a field
- Solving overdetermined linear systems in GF(2)
- Proof of normal basis theorem for finite fields
- Field $\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[3]7+2i$
- Subfield of a finite field with prime characteristic
- Rank of a Polynomial function over Finite Fields
- Finite fields of order 8 and isomorphism
- Finding bases to GF($2^m$) over GF($2$)
- How to arrange $p-1$ non-zero elements into $A$ groups of $B$ where $p$ is a prime number
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Hint: Consider the function $\phi:F\rightarrow F$ by $\phi(x)=x^p$. Under what circumstances is this an automorphism? (It is called the Frobenius automorphism.)
If you can find the conditions under which $\phi$ is an automorphism, the next step is to determine when it is a non-identity automorphism. To do this, think about its order in the automorphism group.
Finally, you've handled almost all of the cases; what's left is to show that a very specific type of field doesn't have non-trivial automorphisms.