1
vote
1answer
41 views

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 ...
0
votes
1answer
23 views

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 ...
1
vote
3answers
47 views

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 $$a^{n-1}\equiv1\text{ mod } n\;\;\;\wedge\;\;\;a^{\frac{n-1}{p}}\not\equiv1\text{ mod }n\;\;\;\forall p\in\mathbb{P}:p\mid n-1$$ ...
0
votes
1answer
18 views

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: ...
0
votes
1answer
21 views

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 ...
1
vote
1answer
67 views

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 ...
0
votes
0answers
36 views

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 ...
4
votes
1answer
69 views

Why is $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$?

I have to show that $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$. I know that $\mathbb{Z}_{51}\cong\mathbb{Z}_3 \times \mathbb{Z}_{17}$ and that $(\mathbb{Z}_p)^*\cong ...
1
vote
2answers
88 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
11
votes
3answers
379 views

Prove that $n^2+n+41$ is prime for $n<40$

Here's a problem that showed up on an exam I took, I'm interested in seeing if there are other ways to approach it. Let $n\in\{0,1,...,39\}$. Prove that $n^2+n+41$ is prime. I shall provide my own ...
1
vote
2answers
43 views

How can you show this relation between primes and roots of unity?

If $p$ is a prime number, how can you show that there are exactly $p^{n-1}(p-1)$ primitive $p^n$-th roots of unity? I am a little stuck on how to begin this proof. Do you need to use orders or ...
1
vote
2answers
55 views

$\mathbb{Q}$ adjoining primes and the sum of root of those primes

I have $p$, $q$ as primes, and I want to show that $\mathbb{Q}(\sqrt{p},\sqrt{q})=\mathbb{Q}(\sqrt{p}+\sqrt{q})$. I was thinking about using inclusion both ways, so what does an element in ...
2
votes
0answers
42 views

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$. So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 ...
0
votes
0answers
45 views

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 ...
1
vote
0answers
45 views

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 $\mathbb{Z} \subset \mathbb{Q}$ ...
1
vote
1answer
86 views

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 ...
1
vote
1answer
97 views

Find solutions to equation (ring/field theory, residue class)

I'm trying to solve this problem: A residue class ring mod $n$ is a field if n is prime. Let $\mathbb{Z}_p$ be a residue ring, p prime. Let $a \in \mathbb{Z}_p$. What are all solutions $x \in ...
3
votes
2answers
109 views

Let $n \in \Bbb N$. Let $p>2$ a prime number. Show that $1^n+2^n+…+(p-1)^n \equiv 0 \pmod {p}$ [duplicate]

This is an exercise in my abstract algebra reader, in the chapter about polynomial rings. Let $n \in \Bbb N$. Let $p>2$ a prime number. And let $n$ not divisble by $p-1$. Show that ...
4
votes
1answer
428 views

An isomorphism that takes Z12 (integers modulo 12 under addition) to Z13* (integers modulo 13 under multiplication)

I'm having a hard time finding an isomorphism that takes the integers in $\mathbb{Z}_{12}$ (those integers modulo 12 under addition) to the integers in $\mathbb{Z}_{13}^{*}$ (those integers modulo 13 ...
2
votes
1answer
284 views

How can I find decompositions in $\mathbb{Z}[\sqrt{d}]$?

Decompositions in $\mathbb{Z}$ In $\mathbb{Z}$ you can find a decomposition of any element $n \in \mathbb{Z}$ by factorization such that $$n = \prod_{p \in \mathbb{P}} p^{v_p(n)}$$ So for a ...
1
vote
2answers
46 views

problem proving this property of congruence and primes

I've been working on this for a few days and I just can't seem to find a good proof for this. Given $a \equiv b\pmod{p_i}$, $i=1,2,3,\dots,n$ and $p_i$ is prime, show that $a \equiv b ...
1
vote
1answer
53 views

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 ...
2
votes
1answer
38 views

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 ...
2
votes
1answer
40 views

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 $n$ ...
8
votes
2answers
105 views

$R$ 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 ...
3
votes
2answers
67 views

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 ...
1
vote
2answers
201 views

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 ...
3
votes
1answer
158 views

Additive primes

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 ...
0
votes
1answer
184 views

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. ...
3
votes
1answer
497 views

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 ...
3
votes
2answers
208 views

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 ...
2
votes
1answer
544 views

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 $\dfrac {n} {abc\ldots k}$, ...
5
votes
1answer
363 views

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 ...
2
votes
2answers
252 views

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 ...
6
votes
1answer
143 views

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 ...
40
votes
2answers
4k views

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 ...
1
vote
1answer
524 views

If the order divides a prime P then the order is P (or 1)

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 $q\equiv 1 \bmod{p}$. ...
9
votes
1answer
878 views

Euler's remarkable prime-producing polynomial and quadratic UFDs

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: ...