Tagged Questions

3
votes
1answer
31 views

Primitve roots and congruences?

Let $p$ be an odd prime. Show that the congruence $x^4$$\equiv -1\text{ (mod }p\text{)}\ $ has a solution if and only if $p$ is of the form $8k+1$. Here is what I did Suppose that $x^4$$\equiv ...
-2
votes
1answer
66 views

Finding a primitive root modulo $13$ [duplicate]

Find a primitive root modulo each of the following integers. a) $13$ My TA said we are not going to go over this. We did not go over the topic. It seems like something good to know though. ...
1
vote
1answer
70 views

Finding a primitive root modulo $11^2$

Find a primitive root modulo each of the following moduli: a) $11^2$ My TA said he is not going to go over this so do not worry about it. He said you can try this if you want but he would ...
1
vote
1answer
59 views

$\mathbb Z_p^*$ is a group.

I'm trying to prove that $\mathbb Z_p^*$ ($p$ prime) is a group using the Fermat's little theorem to show that every element is invertible. Thus using the Fermat's little theorem, for each $a\in ...
5
votes
2answers
69 views

Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$

I'm trying to prove that Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$ (p prime). I know that Aut $\mathbb Z_p$ has $p-1$ elements because $\mathbb Z_p$ has $p-1$ possiblities of generators, so intuitively ...
0
votes
2answers
36 views

Trying to prove $A=\{2, 4, 8,…,2^k\}$ is closed under multiplication in $\mathbb{Z}/(2^{k+1}-2)\mathbb{Z}$

I've been investigating as part of a project the structure of integers modulo $n$ ($\mathbb{Z}$/n$\mathbb{Z}$) under multiplication. One aspect I'm looking at is, for any natural number $k$, finding a ...
1
vote
0answers
74 views

Order of kernel of a homomorphism

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
4
votes
4answers
61 views

What prerequisite knowledge is needed to understand “Multiplicative group of integers modulo n”

I want to self teach myself Multiplicative group of integers modulo n since it's a foundation in cryptography, IT Security, and Microsoft's UProve technology. When I go to the Wikipedia page I am ...
0
votes
3answers
109 views

Show that a map is well-defined and homomorphism

Define a function $f_n:\mathbb{Z}_m \to \mathbb{Z}_m$ as a map $\bar{a}\mapsto n\cdot \bar{a}$. Show that it is both well-defined and a group homomorphism. For the well-defined part, I know that I ...
1
vote
3answers
99 views

Question about rings and modulo multiplication tables

Why is 2=0 in K for part a)? Also I don't understand what part b) is asking you to do - what does it mean by alpha = [X], so M=etc. Could someone please explain the question and the solutions to part ...
1
vote
2answers
101 views

Modular Arithmetic and the Order of the Monster

I am trying to verify directly using modular arithmetic that $|M|/(40*41) \equiv 1$ mod $41$, where $M$ is the monster group. I am not really sure how to approach this type of problem, i.e. one ...
1
vote
1answer
255 views

Solving a polynomial modulo an integer

Say I have a polynomial $F$ of degree $n$ with coefficients in $Z_m$ and I wish to find $x$ such that $F(x)=0$ (mod $m$). For instance if $F(x)=x^{2}-a$ the solution would be the modulo $m$ squareroot ...
2
votes
4answers
353 views

Checking if a set is closed under modular multiplication

I have to check if the set {1, 3, 7, 9, 11, 13, 17, 19} forms a group under multiplication modulo 20, and the only idea that I've had so far is: attempt brute force on all the elements (multiplying ...
3
votes
0answers
112 views

Number of elements in projective special linear group over $\mathbb{Z}/n\mathbb{Z}$

While reading a paper about the modular group $\Gamma = PSL_2(\mathbb{Z})$, I came upon the following sentence ($\Gamma(N)$ is the kernel of the canonical map $PSL_2(\mathbb{Z}) \rightarrow ...
4
votes
1answer
185 views

Fermat's little theorem for $n=3$

for $N > 0$, I'm trying to show Fermat's little theorem, for $3$ using the orbit stabilizer theorem: $N^3 - N$ an element of $3\mathbb{Z}\ (3 \mod \mathbb{Z})$ Pf/ we can break it down into ...
1
vote
1answer
245 views

Breaking RSA in a special case

This is a part of homework assignment, and I am stuck. The RSA signature is being calculated using Chinese Remainder theorem technique. Find the detailed description here. Public and private keys are ...