3
votes
2answers
151 views

Number theory problem.Primes modules.

If $$a^p\equiv b^p \pmod p$$ where $p$ is prime prove that $$a^p\equiv b^p \pmod{p^2}$$ that problem was at my exam today on number theory and i just didnt have a clear mind to solve it.Although i ...
1
vote
1answer
14 views

Definition for a distance function over a residue class ring

I'm searching for a reasonable definition of a distance function $$d:\mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}\to\mathbb{N}_0$$ which satisfies $d(\overline{n-1},0)=1$ ...
0
votes
1answer
40 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
-1
votes
1answer
28 views

Congruents and modulo question- counterexample

I have tired an couple and none seem to be working
1
vote
2answers
42 views

Property of modulo division

I wanted to check if it is true, that $$a^{3b} \pmod n = (a^{b} \pmod n)^{3}\ ?$$ For example when $a = 2, b = 4, n = 5$ I have that $2^{12} \mod 5 = 1$ and $(2^4 \mod 5)^3 = 1$ Is that always true, ...
0
votes
2answers
38 views

Property of Modular arithmetic

If I know that $$g^a \neq 1 \mod b$$ is that always true that if I will take a positive integer $c$ and count $(g^a)^c$, then $$(g^a)^c \neq 1 \mod b$$?
0
votes
0answers
8 views

cracking a linear congruential generator

so i have a problem where i am asked to find a and c where i am given m=97. there are also multiple elements in the sequence given,x1 = 93, x2 = 87, x3 = 29, x4 = 18, x5 = 41. i set up all the ...
0
votes
1answer
33 views

Why is $U(12) = U_{4} (12) ~ U_3(12)?$

Why is $U(12) = U_{4} (2)~ U_3(12)$ Attempt: Any subgroup $U_k(n) = \{x \in U(n)~~|~~x \mod k=1 , k ~|~n \}$ Hence : if $U(12) =\{1,5,7,11\}$ then : $U_4(12) =\{1,5\}$ and $U_3(12)=\{1,7\}$ We see ...
1
vote
1answer
24 views

Factoring in terms of Irreducibles

Factor the polynomial $x^5 + 2x^3 + 3x^2 + 1$ as a product of irreducible polynomials in $\mathbb{Z}_5[x]$. My thoughts: I know what the definition of an irreducible function is but as far as methods ...
0
votes
1answer
47 views

Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
4
votes
2answers
63 views

Why can't we have an $y$ such that $xy\equiv 1\; (mod\; n)$ when $n$ is not prime?

I'm reading Avner's Fearless Symmetry: Here he says that we can only have the cancelation law if the modulus is prime: I got curious with the statement and then I kept reading the chapter: ...
0
votes
1answer
32 views

Solving a system of modular equatios

Edit: I can't actually see how Chinese remainder theorem works here, if we had only $x$ on the left of each equation I can see how I could work it, but we don't. I can't seem to reduce it down to just ...
3
votes
3answers
42 views

If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
0
votes
0answers
10 views

Preservation of a map

There is a map from Z(mod12) to Z(mod4) defined by f(x)=3x. The thought I had was this. Say you have [a],[b] that are in Z(mod12). Would f([a][b])=f([a])f([b])? So you basically view this as a ...
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 ...
0
votes
1answer
28 views

GCD of polynomials

In $\mathbb{Z}/5\mathbb{Z}[x]$ use the Euclidean Algorithm to find the GCD of $x^4+x^2$ and $x^4+x^3+3x$. My thoughts: I am getting to the point where I need to use a fraction to get further in ...
0
votes
0answers
17 views

Question regarding a surjective function

The function $f :\mathbb{Z}_{12} \longrightarrow \mathbb{Z}_4$ is defined by $f(x)=3x$. I am asked given any $a,b \in \mathbb{Z}_{12}$ if preservation of addition and multiplication occur. I proved ...
0
votes
0answers
54 views

Modular arithmatic

Suppose that f : Zmod12 -> Zmod4 is defined by f [x] mod 12 = [3x]mod 4 where the subscript indicates the appropriate modular arithmetic. (A) IS f surjective? (B) is f injective? (C) let [a], [b] be ...
10
votes
2answers
209 views

Are integers mod n a unique factorization domain?

I am trying to learn abstract algebra from scratch, jolly stuff, but in the process of doing so this puzzles me: Having a ring of integers mod n, where $n=pq$ is composite, as I understand we have ...
0
votes
1answer
25 views

Proving that $mx \equiv 0 \pmod n$ has $\gcd(m, n)$ solutions in the interval $[0, n-1]$

I wish to prove, using my own intuition, that there are $\gcd(m, n)$ group homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$. I have reduced(?!) the problem to proving that there are $\gcd(m, n)$ ...
1
vote
3answers
47 views

Multiplying modulo 20

For a positive integer $n \ge 2$, define $U(n)= \{k \in \Bbb Z_n $| gcd$(k,n)=1\}$. Then $U(n)$ is a group under multiplication modulo $n$. Find the order of $U(20)$. Is it possible to generate ...
3
votes
5answers
300 views

Modular Arithmetic - Are we allowed to distribute the Modularity?

Assume I have a problem such as "Prove that $\displaystyle103^{53} + 53^{103}$ is divisible by $39$." This would mean I wanted to prove that $\displaystyle103^{53} + 53^{103}\equiv0\pmod{39}$. My ...
0
votes
1answer
52 views

Show that the mod p map is a ring homomorphism

Let p be a prime and let (mod $p$)$ : Z[x] \mapsto Z_p[x]$ be the mod-p map which sends any polynomial... $f(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n \in Z[x]$ ... to the polynomial... $f(x)$(mod ...
1
vote
2answers
42 views

Modular arithmetic question related to the fundamental theorem

Somewhat of an unusual homework problem that my professor assigned that I can't wrap my head around. We are only considering the positive numbers congruent to 1(mod 4), that is, other numbers do not ...
1
vote
0answers
47 views

Group of invertible elements, isomorphic to $Z_4$?

The group $f(8)$ of invertible elements in the ring $Z_{10}$ has four elements, $f(10) = \{ [1,], [3], [7], [9]\}$. Is this group isomorphic to $Z_4$ or to the symmetry group of the rectangle? ...
2
votes
3answers
50 views

modular arithmetic proof

Suppose $x$, $y$, and $z$ are integers and $x= 3y^2 -z^2$. Prove that $x\not\equiv1\mod4$. My thoughts: So I am not sure the route that can prove this. I am trying to just use the simple stuff to ...
1
vote
1answer
55 views

Zn modular tables , invertible elements and zero/nonzero divisors

Based on Zn, with n <= 10, make a guess about which elements in Zn are invertible and which are nonzero divisors. Does your guess imply that every nonzero element is either invertible or a zero ...
2
votes
2answers
107 views

Prove that $ax=0$ has a nonzero solution in $\mathbb{Z}_n$ $\Leftrightarrow$ $ax=1$ has no solution.

Let $a\neq 0$ in $\mathbb{Z}_n$. Prove that $ax=0$ has a nonzero solution in $\mathbb{Z}_n$ if and only if $ax=1$ has no solution. $\textbf{My proof (just one way)}$: ($\Rightarrow$) Suppose $a\neq ...
0
votes
1answer
50 views

Solving an equation with modular arithmtics

Consider a machine that operates on the set of real numbers using the following equation: O = X × [(I + Y) mod L] − Z, where I is the input and O is the output of this machine. L is known and it is ...
0
votes
1answer
54 views

Modular arithmetic multiplication table exam question

I am stuck on this past paper question, and looking at the solution didn't seem to give me any hints either. I'm stuck on part b but I'll write the first part out anyways. Let $K =\mathbb Z_2$. Let ...
1
vote
2answers
53 views

Show that $M$ is a submonoid of the group of $2\times 2$ matrices of integers mod $13$.

Define $$M=\left\{\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\in(\Bbb Z/13)_{22}:a_{12}=0\right\}\;.$$ (This is the set of $2\times 2$ matrices with integers mod $13$ with the ...
0
votes
1answer
52 views

Finding subgroups via modulo arithmetic [closed]

Consider the following numbers : $1, 3, 7, 9, 11, 13, 17, 19$. In terms of modulo $20$ how many subgroups are there? For example, $3(7) = 21 = 1$ mod $20$. In particular I want to find the ...
0
votes
1answer
65 views

A nice group isomorphism

Show that $$k(\mathbb{Z}/n\mathbb{Z})\cong (\gcd(n,k)\mathbb{Z})/n\mathbb{Z}.$$ I want to see as many as possible proofs of this nice fact.
-1
votes
1answer
80 views

Find unity of ring.

The ring {0, 2, 4, 6, 8} under addition and multiplication modulo 10 has a unity. What is that unity and how do we find it?
1
vote
2answers
48 views

Abstract Algebra: Struggles with rings

This is a part of the group of practice problem I've been working on and I'm just lost. I'm really struggling when it comes to these ring problems. Anybody who could lay out an outline for this ...
1
vote
1answer
59 views

Prove $f(x)=9x^2-5y^2-34$ has no integral roots

Prove $f(x)=9x^2-5y^2-34$ has no integral roots. I have tried working this mod 2, 3, 4, 5, and 17, and some random others, to no avail. It is for a graduate course, so I am thinking maybe I tried to ...
4
votes
2answers
65 views

How to prove that $a^{2^{n-2}} \equiv 1 \pmod{2^n}$?

Let $a$ be an odd integer and $n$ an integer such that $n\ge 3$. 1) I want to show that $a^{2^{n-2}} \equiv 1 \pmod{2^n}$ 2) Then I want to show that $(\mathbb Z/{2^n\mathbb Z})^*$, the ...
0
votes
1answer
42 views

Chinese remainder theorem and order

How do I find out which elements of the monoid (Z/161Z, *, 1) are not invertable? I'm trying to find the group of units but I can't really grasp which elements are invertible and which aren't.
2
votes
1answer
176 views

Prove that the mapping $U(16)$ to itself by $x \rightarrow x^3$ is an automorphism

Prove that the mapping $U(16) = \{{1,3,5,7,9,11,13,15}\}$ to itself by $x \rightarrow x^3$ is an automorphism. What about $x \rightarrow x^5$ and $x \rightarrow x^7$? any generalization? So far i ...
1
vote
0answers
233 views

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them.

How many ring homomorphism are there from $\mathbb{Z} / 12\mathbb{Z}$ to $\mathbb{Z} / 20\mathbb{Z}$? Find the image and kernel of them. The above is the question, this is my attempt at an ...
1
vote
2answers
51 views

Find a positive integer $x$ less than $105$ satisfying the following simultaneous congruence equations.

$$x=2 mod 3$$ $$x=3 mod 5$$ $$x=4 mod 7$$ I have only learnt modulo for 2 weeks so far... really basic theorems. My attempt using definitions of modulo From Equation 1, $3a=x-2 ...
0
votes
1answer
112 views

Is a congruence (equivalence) class modulo n a group?

I know that the set of all equivalence classes Z/nZ is a group (with identity element the equivalence class [0], inverse element -[a]=[n-a]=[-a], etc.). However, is a single equivalence class modulo ...
2
votes
1answer
181 views

Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups

Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$. A. Find integer $n$ such that $U(n)$ contains ...
3
votes
1answer
105 views

Natural Representation of Factor Group $G/H$

Let $G$ be the positive reals under multiplication and let $H$ be numbers $2^i$ where $i \in \mathbb{Z}$. a) Show H is a subgroup of G b) Show H is a normal subgroup of G Those two are no problem, ...
4
votes
1answer
425 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 ...
1
vote
3answers
215 views

Show $f(X)=a_nX^n+\cdots+a_1X+a_0$ has degree $n$ modulo $N$, $f(a)\equiv 0$ (mod $N$) then $f(X) \equiv (X-a)g(X) $(mod $N$)

In Niels Lauritzen, Concrete Abstract Algebra, I'm having trouble showing the following: The problem starts out like this: $f(X)=a_nX^n+\cdots+a_1X+a_0, a_i \in \mathbb Z, n \in \mathbb N$ Part ...
4
votes
2answers
149 views

Show that $c = \max(a, b)$ on $\mathbb{Z}_2$ is not a binary operation

Let $*: \mathbb{Z}_2\times\mathbb{Z}_2 \to \mathbb{Z}_2$, be defined as $[a] * [b] = [c]$, where $c = \max\{a, b\}$, for all $[a], [b] \in \mathbb{Z}_2$. Prove that $*$ is not a binary operation on ...
1
vote
1answer
461 views

Do (a+b)mod n=a'+b' as same as (a+b) mod n= (a'+b') mod n?

The question is:Let n be a fixed postive integer greater than 1.If a mod n= a' and b mod n =b',prove that (a + b) mod n= (a' + b') mod n and (ab) mod n= (a'b') mod n. My answer so far is:From ...
1
vote
2answers
127 views

Prove: $f(x)^{p^k}\equiv f\left(x^{p^k}\right)\bmod p$

$p$ is a prime number, $k$ is an positive integer, and $f\in\Bbb Z[x]$. Prove: $f(x)^{p^k}\equiv f\left(x^{p^k}\right)\bmod p$
3
votes
0answers
87 views

Equivalence classes of triplets satisfying $x^2+y^2+z^2=0$ over $\mathbb{F}_p$

The affirmative answer to this question illustrates that the equation $$x^2+y^2+z^2=0$$ has $p^2-1$ nontrivial solutions over $\mathbb{F}_p$ (solutions that are not $(0,0,0)$). If $(x,y,z)$ is a ...