# Tagged Questions

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### What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
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### Non-obvious deduction regarding conjugates in $\text{GL}_2(\mathbb{F}_p)$

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2,$ the set of $2$-vectors with entries in $\mathbb{F}_p$, by matrix multiplication. $[\,$Prove that$\,]$ for any ...
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### Ring theory Algebra

Let F be a field of $8$ elements and A=set of all x belongs to F such that $x^7=1$ and $x^k \neq 1$ for all $k < 7$. then the number of elements in A is
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### Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F}$

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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### $F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if ...
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### Simpler method to show $k(x)$ is NOT a primitive irreducible polynomial?

Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive. Ok, I showed that $k(x)$ is irreducible since it it a quadratic and ...
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### Group-Isomorphism problem

I want to find an group-isomorphism $$\psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times$$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
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### elementary abelian groups and finite fields

in group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order $p$, where $p$ is a prime; it is a particular kind of $p$-group. now suppose that we ...
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### automorphism of fields

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field ...
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### Number of elements which are cubes/higher powers in a finite field.

This question is a slight generalization of This Question. How many elements are there in a finite field of order $q$ which are : Squares. Cubes. Higher powers. I mean : How many elements are ...
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### For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
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### example of Frobenius endomorphisms that is not automorphism

Let $F$ be a field of characteristic $p$ and $F$ has infinitely many elements. Prove that the map $\sigma: \alpha\mapsto \alpha^p$ is a field endomorphism but not necessarily automorphism. How to ...
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### If $F$ is a finite field then $|F|=p^n$ for prime $p$ and some integer $n$ [duplicate]

I need some help proving the following: If $F$ is a finite field then $|F|=p^n$ for prime $p$ and some integer $n$. By contradiction, suppose $|F| =pq$ for primes $p$ and $q$. Then by Cauchy's ...
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### Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
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### Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\not|n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
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### Formal derivatives over finite fields.

I am slightly confused about what formal derivatives over finite fields mean. Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$. By checking each element of $\mathbb{F}_7$ we easily see that this ...
The group of units of the rings $\text{GF}(p^r)$ and $\mathbb{Z}/p^r\mathbb{Z}$ are both cyclic (except for the exception of prime powers are not cyclic when $p=2$ and $r\ge 3$). This is a strong ...
### Extracting the value of $y$ from $x$ in an elliptic curve over a finite field
Given an elliptic curve $y^2 = x^3 + ax + b$ over a finite field $\mathbf{F}_p$, how can I retrieve the value of $y$ given the value of $x$? My knowledge in this area is quite limited, so I ...