0
votes
1answer
15 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
1
vote
1answer
21 views

Extensions of nonarchimedean valuations

This is a question from Janusz 'Algebraic Number Theory'. Let $R$ be a DVR with maximal ideal $\mathfrak p=\pi R$. Let $K$ be the quotient field of $R$ and $\mid\cdot\mid_{\mathfrak p}$ the ...
1
vote
3answers
65 views

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that ...
5
votes
0answers
36 views

Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight. Let $k$ be a $p$-adic or number field, and let ...
3
votes
1answer
34 views

Showing two numbers are relatively prime in number fields

Solve $x^3-2=y^2$ in integers. The standard way to solve this problem is to consider the arithmetic of the ring of algebraic integers $\mathcal{O}_{\mathbb{Q}(\sqrt{-2})}$ and to show that ...
23
votes
1answer
233 views

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$? I think there are no such polynomials, but how to prove?
5
votes
3answers
44 views

Let $\mathbb Q\le K$ finite extension of fields $\alpha\in K$. Then $N(\alpha)=\pm1\Leftrightarrow\alpha$ is a unit

Let $\mathbb Q\le K$ (where $K\le\mathbb C$) be a finite extension (say $|K:\mathbb Q|=n$) of fields and let $\alpha\in K$ be an algebraic integer, i.e. $\alpha$ is a root of a monic polynomial over ...
5
votes
1answer
98 views

Units of $\mathbb{Z}[\sqrt[4]2]$

How would one compute the units in $\mathbb{Z}[\sqrt[4]2]$? According to one source, it can be shown that the fundamental units are $1 + \sqrt[4]2$ and $1 + \sqrt{2}$, but it does not specify the ...
3
votes
1answer
66 views

Determine Units of a Ring $\mathbb{Z}[\alpha]$

I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
3
votes
1answer
44 views

tensor product writing of $\mathbb{C} / (2\pi i)^n \mathbb{Q}$

Can someone explain to me why $\mathbb{C}/(2\pi i)^n \mathbb{Q}$ is isomorphic to the tensor product $(2\pi i)^{n-1} \mathbb{C}^{\times} \otimes_{\mathbb{Z}} \mathbb{Q}$? I don't understand well the ...
5
votes
2answers
95 views

The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.

I have to study the prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
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 , ...
0
votes
1answer
31 views

Modulation and translation properties of DFT

Consider the discrete fourier transform over a finite field $GF(q)$. Let also $\omega$$\in$$GF(q)$ be an element of order $n$ and which is an $n$-th root of unity. Definition 1. Let $v$ = ($v_0$, ...
1
vote
2answers
47 views

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y [duplicate]

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y ...
3
votes
2answers
95 views

Is there any usage of this sum formula?

Let $\phi$ be the Euler phi function, then $$n=\sum_{d|n}\phi(d)$$ It is one of the most famous formulas in number theory. It can be generalized by using groups. Let $G$ be a group of order $n$ and ...
1
vote
1answer
55 views

Number of solutions of some congruence equations.

How many $[u]\in(\mathbf{Z}/ab\mathbf{Z})^\ast$ satisfy the equations $u\equiv 1 \bmod \ a$, $u\equiv 1 \bmod \ b$? I somehow believe that the answer might be $(a,b)$. Is this actually true? Is the ...
2
votes
1answer
74 views

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
1
vote
1answer
21 views

Does $a^{\phi(p^k)/2} \equiv -1 \pmod {p^k}$ imply $a$ is a primitive root modulo $p$, where $p$ is an odd prime and $k$ an integer $\ge 2$?

Suppose $a$ is a primitive root $\pmod p$. Does $$a^{\phi(p^k)/2} \equiv -1 \pmod {p^k}$$ imply $a$ is a primitive root modulo $p^k$, where $p$ is an odd prime and $k$ an integer $\ge 2$ ? I've been ...
1
vote
1answer
41 views

Prove $n>1$ is prime $\iff \alpha^{n-1} = 1 \ \forall \alpha \in \mathbb Z_n\setminus\{[0]\}$.

Prove $n>1$ is prime $\iff \alpha^{n-1} = 1 \ \forall \alpha \in \mathbb Z_n\setminus \{[0]\}$. I have proven $\Rightarrow$ which is an immediate consequence of Euler's theorem, however I ...
2
votes
3answers
115 views

Questions for first year students at the University. [closed]

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is ...
0
votes
2answers
83 views

Closure of Integers under multiplication and rational exponentiation

What is the closure of the Integers under a finite number of multiplications and rational exponentiations? For example, $3^{1/2}$, $i = -1^{1/2}$, and $\frac{-1+i \sqrt(3)}{2} = 1^{1/3}$ all in this ...
2
votes
1answer
49 views

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$?

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$ ? And why does $p_i^{\alpha_i} \mid \text{ord}_n(a)$ ? I see that ...
5
votes
1answer
56 views

Using Plancherel's Theorem to Prove the Gauss Sum

I'm interested in proving the following: Where $p$ is an odd prime and $z$ is a primitive $p$th root of unity, we let $Q(p)=\sum^{p−1}_{k=0}z^{k^2}$. Prove: $|Q(p)|=\sqrt{p}$. Specifically, I want ...
1
vote
3answers
104 views

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$.

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$, where $n$ has a primitive root. I have proven the following theorem ...
3
votes
1answer
96 views

Euclid's Lemma for Euclidean Ring.

Question: If $R$ is a euclidean ring and $\pi\in R$ is irreducible, prove that $\pi\mid\alpha\beta$ implies $\pi\mid\alpha$ or $\pi\mid\beta$. A solution is to prove all euclidean rings are PIDs, ...
2
votes
0answers
30 views

automorphism of $\Bbb{Q_p}$closed [duplicate]

How to show that there does not exist an automorphism of $\Bbb{Q_p}$ except identity. Please help. Is any automorphism of $\Bbb{Q_p}$ already continuous.?
1
vote
1answer
84 views

Question on complete discrete valuation field.

Let $F$ be a complete discrete valuation field and $f(X) = X^n + a_{n-1}X^{n-1} +\cdots+ a_0$ is an irreducible polynomial over $F$. How to show that a) $ v(a_0) > 0$ implies $v(a_i) > 0$ for ...
3
votes
2answers
60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
2
votes
2answers
125 views

Silly mistake in this number theory book

My question is very easily to be solved (at least I hope so) I think this book has a mistake: When I calculate I get $b_3\equiv -2 (\mod{2})$ which implies $b_3=0$, am I right? Another question, ...
0
votes
0answers
22 views

Why does this theorem provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution?

Why does the following theorem both provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution ? I see that $\gcd((ad-bc),m) = 1$ is a ...
6
votes
1answer
75 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
2
votes
1answer
84 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
0
votes
0answers
15 views

$m^2 = \prod_{i=1}^k p_i^{2e_i} \equiv 1 + \sum_{i=1}^k e_i (p_i^2-1) (\mod 64)$ imply $(m^2 - 1)/8 \equiv \sum_{i=1}^k e_i (p_i^2-1)/8 (\mod 2)$?

Let $m$ be an odd positive integer with canonical decomposition $\prod_{i=1}^k p_i^{e_i}$. I know $$m^2 = \prod_{i=1}^k p_i^{2e_i} \equiv \prod_{i=1}^k [1+e_i(p_i^2-1) \equiv 1 + \sum_{i=1}^k e_i ...
1
vote
1answer
35 views

What can we say about this quantity?

Let $\phi(n)$ be the Euler phi-function. If $a>1$ is an integer, then what is the remainder when $\phi(a^n - 1)$ is divided by $n$ in accordance with the Euclidean algorithm?
1
vote
1answer
10 views

Gauss' Lemma: $r_1, \ldots, r_k, p - s_1, \ldots, p-s_{\nu}$ are all incongruent where $r_i, s_j$ are least residues.

I'm having trouble understanding a step in the below proof of Gauss' Lemma. I see that $r_1, \ldots, r_k, p - s_1, \ldots, p-s_{\nu}$ are all less than $p/2$ and it follows that $r_1,\ldots, r_k$ ...
1
vote
1answer
43 views

Factorization of an ideal in a number field

The notes I read gives following technique to factor an ideal in a number field without explanation. Can anyone explain how this technique works? To factor the ideal $(2)$ in $\mathbb{Z}[\sqrt{-5}]$, ...
1
vote
3answers
51 views

Roots in $\mathbb{Z}_m$

Suppose we define $S_m\subseteq \mathbb{Z}_m[x]$ to be the set of all polynomials $f$ that has roots everywhere in $\mathbb{Z}_m$, i.e. $f(a)=0$ for all $a\in \mathbb{Z}_m$. It is immediate by FLT ...
1
vote
1answer
53 views

Proof of an alternative form of Fermat-Euler's theorem.

I want to know a proof of an alternative form of Fermat-Euler's theorem $$a^{\phi (n) +1} \equiv a (mod \; n)$$ when a and n are not relatively prime. I searched some number theory books and a ...
2
votes
1answer
34 views

Diophantine Equations problem 2

Find all the solutions to the Diophantine equation x^2+y^2=2(z^2) .I do not have alot of expirience on Diophantine equations and i do not know how to approximate them.I can see that the tripples of ...
2
votes
2answers
31 views

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer).

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer). My attempt:If $p$ is a prime, then $U_p=${$[x]|1\leq x<p$} is cyclic.
1
vote
1answer
79 views

Prove that $(a, b)$ is prime in $\mathbb{H}$ if $||(a, b)||$ is prime in $\mathbb{N}$

Numbers in $H$ are ordered pairs of integers, i.e. $(a,b) \in \mathbb{H}$ if $a \in \mathbb{Z}$ and $b\in\mathbb{Z}$ Multiplication is defined as $(a, b)$ x $(c, d) := (ac-5bd , ad+bc)$ For any ...
0
votes
2answers
39 views

Prime pairs ($p$, $q$) such that $p|q^m - 1$ for some integer $m$

Let $p$ and $q$ be two different prime numbers. Is it true that there exist an integer $m$ such that $p | q^m - 1$? If no, what family of prime pairs are known to have the above property?
1
vote
1answer
50 views

What is known about the ramification index of ramified primes in an arbitrary cyclotomic extension of $\mathbb{Q}$

Let $\zeta$ be a primitive $m$th root of unity, and $L = \mathbb{Q}(\zeta)$. Then $B = \mathbb{Z}[\zeta]$ is the integral closure of $\mathbb{Z}$ in $L$. If $P$ is a prime ideal of $B$ and ...
0
votes
3answers
50 views

Prove that n divides $\phi(a^n-1)$, where $\phi$ is Euler's $\phi$-function.

Let a, n be positive integers. Prove that n divides $\phi(a^n-1)$, where $\phi$ is Euler's $\phi$-function. I know this problem can be done using number theory approaches, however I am rusty on those ...
0
votes
2answers
108 views

$\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?

(This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic) I was trying to solve this: Let $G$ be a finite abelian group of order $n$ for which the ...
0
votes
1answer
47 views

Automorphisms and even permutations

One question, two parts... (a) Consider the group $(Z_n, + \mod{n})$. If $n$ is an odd prime number, determine (with proofs) if all automorphisms of $(Z_n, + \mod{n})$ are even permutations of $Z_n$. ...
1
vote
4answers
65 views

If p is a prime number of the form $4n+3$, show that we cannot solve $x^2\equiv -1\mod p$

Hint: Use Fermat's Theorem that $a^{p-1}\equiv 1\mod p$ if $p \nmid a$. (I have no idea, but something in group theory should help)
0
votes
1answer
41 views

How to prove that multiplication distributes over addition in a certain number system H

Numbers in $H$ are ordered pairs of integers, i.e. $(a,b) \in \mathbb{H}$ if $a \in \mathbb{Z}$ and $b\in\mathbb{Z}$, with addition and multiplication defined by $$(a,b)+(c,d) := (a+c,b+d),\\ ...
1
vote
2answers
133 views

The Proof of Wilson's Theorem using the auxiliary multiplicative modulous group [duplicate]

(self answered question, thanks for the hints Derek Holt provided:-)) problem 18,section 4 chapter 2 in Herstein's abstract algebra: Using the results of Problem 15 and 16,prove that if p is an ...
3
votes
0answers
36 views

Elements of a Dedekind domain can be chosen to have valuation $1$ with respect to one prime, $0$ everywhere else

I noticed this is true for $\mathbb{Z}$, but I was wondering whether it was true in general. Let $R$ be a Dedekind domain and $P_1, ... , P_s$ maximal ideals. The localized ring $R_{P_i}$ is a ...