# Tagged Questions

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### How many $\overline{a}\in\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$ pass the Fermat and Miller-Rabin primability tests?

Let $$\text{F}_{91}:=\left\{\overline{a}\in\left(\mathbb{Z}/n\mathbb{Z}\right)^\times:91\text { passes the Fermat primality test to base }a\right\}$$ and ...
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### Showing $f\in\mathbb{F}_{p^d}[X]:f'=0\Rightarrow\exists g\in\mathbb{F}_{p^d}[X]:f=g^p$

Let $\mathbb{F}_{p^d}$ denote the final field with $p^d$ elements and $\mathbb{F}_{p^d}[X]$ denote the polynomial ring in $X$ over $\mathbb{F}_{p^d}$. How can we show ...
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### If there is an $a\in\mathbb{Z}$ with $a^{n-1}\equiv 1\mod n$ but $a^{\frac{n-1}p}\not\equiv 1$ for all primes $p\mid n-1$, then $n$ is a prime

Let $n\in\mathbb{N}$ with $n\ge 3$ and $a\in\mathbb{Z}$ such that $$a^{n-1}\equiv1\text{ mod } n\;\;\;\wedge\;\;\;a^{\frac{n-1}{p}}\not\equiv1\text{ mod }n\;\;\;\forall p\in\mathbb{P}:p\mid n-1$$ ...
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### Relationship between the Carmichael function and Euler's totient function

Let $\lambda$ denote the Carmichael function and $\varphi$ Euler's totient function. Furthermore, let $p$ denote any prime number and $k\in\mathbb{N}$. The wikipedia article about $\lambda$ states: ...
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### Lemma about a prime ideal in a commutative ring with identity

I am trying to prove the Cyclotomic polynomial is irreducible over $\mathbb{Q}[x]$ for any prime $p$ using Eisenstein's Criterion. However, I would like to be more specific and prove the following ...
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### Prove the center of $G$ cannot have order $p^{n-1}$

Let $p$ be a prime, let $n>2$ be an integer, and let $G$ be a nonabelian group of order $p^n$. Prove the center of G cannot have order $p^{n-1}$. Honestly I have no idea where to start. Perhaps ...
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### Primes probability for $2^{2(ak+b)}-3$

I'm working on the following problem: If $x$ is a prime and of the form $ak+b$, is there a possibility to check, whenever $2^{2x}-3$ could be a prime or not, without calculating it or extracting ...
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### Primality of a binary number

I'm interested in finding out if there is a way to detect if a given binary number is prime. I do not which to convert it to base 10 then use some primality test. Does anyone know it there's a pattern ...
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### Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
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### Prime Splits Completely in Every Intermediate Field

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits ...
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### If $p$ is irreducible and $p \not \mid a$, then $\text{gcd}(p,a)=\pm 1$.

I will be taking a Rings and Fields course in the Fall, so I figured I would read ahead in the textbook (A First Course in Abstract Algebra, by Anderson and Feil) to prepare. Recall the following ...
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### Regularity of matrix with coefficients from GF$(p)$

I have matrix $A$ (its size is $n \times n$) with coefficients from GF($p$), where $p$ is prime. How can be proven that this matrix has all lines linearly independent iff det$(A)\neq 0$(mod $p$). I ...
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### Error in understanding the theorem about the invertibility of an element(coset) of a quotient ring

There's a theorem in Abstract Algebra which states that: An element of a quotient ring $\mathbb{Z}/\langle n \rangle$ or $\mathbb{Z_n}$ that is a coset $\overline{a}$ is invertible iff $a$ and $n$ ...
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### $R$ with an upper bound for degrees of irreducibles in $R[x]$

One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as If $f\in\mathbb{R}[x]$ has degree larger ...
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### Why doesn't work Integer factorization for fields?

I try to unterstand, why the Integer factorization is only working for rings and not for fields. My first idea was, that you don't have a uniqueness quantification for prime "numbers" in fields. Is ...
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### What is the least residue mod $N = 95$ of $3^{1.1 \cdot 10^{43}}$?

This is a practice problem. Since $5 \cdot 19$ are prime factors of $95$ I tried to break it into two congruence equations and use CRT, but I can't seem to work this out. By Fermat's little theorem we ...
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The product of two integers is always an integer. However, the quotient of two integers is not always an integer. This simple fact leads directly to concepts such as "divisibility", "divisors" and ...
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### Prime number in a grid for an identity matrix??

I was reading this Transversal of Primes, and the solution shown for an 11x11 grid. Made me think of an identity matrix. First, have each $a_{ij}$ be either 1 for a prime number or 0 otherwise. ...
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### Factorising into Gaussian primes

I'm trying to factorise the Gaussian integer $z =11 - 3i$ into Gaussian primes. Taking the Euclidean norm on $z$ is $\nu (z) = 130$ which factorises into $2 \times 5 \times 13$ and so I'm assuming I ...
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### Find $X$, $a$ : ALL prime factors of $(X^a - 1)/(X - 1) < X$

where $X$ is an odd prime, and $a$ is an odd integer. For example, let $X = 37$, $a = 3$, we get $$\frac{37^3-1}{36} = 3 \times 7 \times 67.$$ When factoring numbers such as this, I've noticed that ...
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### primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
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### Any power of a prime-length cycle is a cycle

Having some doubts proving exercise statement from Pinter's book. Here's quote: Let $\alpha$ be a cycle of length $s$, say $\alpha = ( a_1, a_2 ... a_s )$. Prove that, if $s$ is a prime ...
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### Set of prime numbers and subrings of the rationals

Let $P$ denote a set of prime numbers and let $R_{P}$ be the set of all rational numbers such that $p$ does not divides the denominator of elements of $R_{P}$ for every $p \in P$. If $R$ is a subring ...
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### Special types of extensions of fields

Let $K$ be a field. Let $p$ be any prime number. Can one always construct an algebraic extension $K_p$ of $K$ with the following properties? (1) If $L$ is a finite extension of $K$ contained in ...
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### The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
I've just come up with this question as I'm studying for a number theory midterm. If $p$ and $q$ are different prime numbers, and it's known that $2^p \equiv 1 \bmod{q}$, then $q\equiv 1 \bmod{p}$. ...
Good example of a polynomial which produces a finite number of primes is: $$x^{2}+x+41$$ which produces primes for every integer $0 \leq x \leq 39$. In a paper H. Stark proves the following result: ...