Let $F$ be a finite field with $p^n$ elements with prime $p > 5$ and $G = S_5$ the symmetric group. Let $l$ be the l.c.m. of the orders of the elements of G and $\theta$ be the primitive $l^{th}$ root of unity over $F$. Define the multiplicative group $$T = \{k\ | \theta \to \theta^k \ \text{ is automorphism of} \ F(\theta) \ \text{over} \ F\}$$ Then what is $|T|$?
2026-03-25 06:03:59.1774418639
Number of elements in the cyclotomic $F$ class.
63 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in GALOIS-THEORY
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- A weird automorphism
- $S_3$ action on the splitting field of $\mathbb{Q}[x]/(x^3 - x - 1)$
- Question about existence of Galois extension
- Prove that K/L is a Galois extension
- discriminant and irreducibility of $x^p - (p+1)x - 1$
- galois group of irreducible monic cubic polynomial
- Proof of normal basis theorem for finite fields
- Regular inverse Galois problem for Q(t)
- When a certain subfield of $\mathbb{C}(x,y^2)$ is Galois
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
Related Questions in SYMMETRIC-GROUPS
- Orbit counting lemma hexagon
- A "Restricted Sudoku" Symmetry Group Question
- Show, by means of an example, that the group of symmetries of a subset X of a Euclidean space is, in general, smaller than Sym(x).
- Prove that $\sigma$ is a power of $\tau$ when they commute $\sigma\tau=\tau\sigma$.
- Proof verification - the only group of order 24 without normal sylow subgroup is $S_4$.
- Symmetry subgroup of a cube
- Subgroup generated by $S$ is $A_5$
- Question about semigroups of permutations
- Symmetry of the tetrahedron as a subgroup of the cube
- Interpretation of wreath products in general and on symmetric groups
Related Questions in ROOTS-OF-UNITY
- On multiplicative and additive properties of cyclotomic polynomials
- Roots of $z^3 + 3iz^2 + 3z + i = 0$?
- Compute the determinant.
- Polygon discriminant sequence
- Is $\sqrt[6]{3} \in \mathbb{Q}(\sqrt[8]{21})$ and/or $\sqrt[4]{5} \in \mathbb{Q}(e^{\frac{2 \pi i}{25}})$?
- How to prove the following identity using complex numbers?
- Why does $\sqrt[4]{-2}=\frac{1+i}{\sqrt[4]{2}}$?
- Square root of a root of unity.
- Rational Trig Solutions for $n\ge 3$
- Solving simultaneous equations using de Moivre's Theorem and Roots of Unity
Related Questions in AUTOMORPHISM-GROUP
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- A weird automorphism
- Confusing step in proof of property of cyclic group automorphisms
- ord$(a) = p, f(a) = a, \forall f : G \to G$ automorphism $\implies |G|$ is not square-free
- Arbitrary automorphism function on Aut(Quaternion Group)?
- writing a computer program in magma that finds a linear code and a specific automorphism group to the code.
- Let $G$ be a group. Show that, for every $a\in G$, the map $\phi_a:G\to G$, defined by $\phi_a(g) := aga^{−1}$ ($g\in G$), is a group automorphism.
- homomorphism from $F^\times \times F^\times$ to Aut$(F)$
- Extension of isomorphism of fields
- Graph with distinct automorphisms but no fixed-point free automorphism
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?