Tagged Questions

1
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1answer
48 views

Localisation and extension of rings

Is $\mathbb{Z}_{(3)}[i,\sqrt{2}]=(\mathbb{Z}[i,\sqrt{2}])_{(3)}$ (where by subscript $(3)$ we mean localisation at the ideal generated by $3$)? Do both of these rings contain elements like $$ ...
4
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1answer
59 views

A question on factorial rings

Is 31 irreducible in the ring $\mathbb{Z}\left[\sqrt{5}\right]=\left\{a+b\sqrt{5}:a,b\in\mathbb{Z}\right\}$ ? And is it prime in $\mathbb{Z}\left[\sqrt{5}\right]$?
4
votes
4answers
95 views

The elements of $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$

I'm really confused with this one... How can I determine the elements of the module $\mathbb{Z}[\sqrt{-5}]/(2,\sqrt{-5}+1)$? Or its cardinality? Does ...
3
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1answer
36 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
0
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1answer
47 views

Ring Isomorphism Proof

Let $p$ be a prime with $p \equiv 1 (\mod 4 )$. I am trying to show that $\mathbb{Z}[X]/(X^2 + 1, p) \cong \mathbb{Z}_p \times \mathbb{Z}_p$ is a ring isomorphism. I am not really sure how to ...
4
votes
2answers
56 views

Direct product of polynomial rings

Let $n = pq$, where $p$ and $q$ are distinct primes. I am trying to show that: $$\mathbb{Z}_n[X] \cong \mathbb{Z}_p[X] \times \mathbb{Z}_q[X].$$ Would it suffice to say that $\rho(np) = ...
0
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1answer
52 views

Smallest Subring

Suppose that $S$ and $T$ are subrings of a ring $R$. Show that their ring-theoretic product $ST$ is a subring of $R$ that contains $S \cup T$, and is the smallest such subring. I understand that $ST$ ...
0
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2answers
34 views

Zero Divisor Rings

Let $R$ be a ring, and let $a,b \in R$ such that $ab \ne 0$. Show that $ab$ is a zero divisor if and only if $a$ is a zero divisor or $b$ is a zero divisor. I understand that a zero divisor is $ab = ...
2
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4answers
94 views

Show that if $c_1 + c_2\sqrt{5}$ divides $n$ in ${\bf{O}}[\sqrt{5}]$, then so does $c_1 - c_2\sqrt{5}$

I have a ring: $${\bf{O}}[\sqrt{5}] = \{c_1 + c_2\sqrt{5}: (c_1 \in \mathbb{Z} \wedge c_2 \in \mathbb{Z}) \lor (c_1 + \frac{1}{2} \in \mathbb{Z} \wedge c_2 + \frac{1}{2} \in \mathbb{Z}) \}.$$ I ...
2
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1answer
51 views

Whether a domain is Dedekind or not

We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
0
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0answers
53 views

Probability that two Gaussian integers are divisible

Let $z = x+ iy \in \mathbb{Z}[i]$ and let $a+ib \in \mathbb{Z}[i]$ with $a^2 + b^2 \equiv 1 \mod{4}$. What is the probability that $a+ib$ divides $x + iy$ in $\mathbb{Z}[i]$? This question would ...
2
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2answers
105 views

Probability that $x \equiv 3 \pmod{4}$

I am working on a number theory project and I am interested in the following statement: What is the probability that an integer $x$ has the property that $|x| \equiv 3 \mod{4}$? This seems ...
3
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4answers
97 views

Non-commutative or commutative ring or subring with $x^2 = 0$

Does there exist a non-commutative or commutative ring or subring $R$ with $x \cdot x = 0$ where $0$ is the zero element of $R$, $\cdot$ is multiplication secondary binary operation, and $x$ is not ...
3
votes
2answers
141 views

Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
2
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2answers
55 views

An ideal, $I$, is maximal iff $R/I$ is a field

The first line of the proof given in my book says that the ideals of $R/I$ are in bijective correspondence with the ideals of $R$ lying between $I$ and $R$. What is the bijection?
1
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2answers
50 views

Integral extensions

Let $p\neq1$ be an integer and let $\beta$ be a root of $x^6-p$. What is the difference, in terms of $\mathbb{Z}$-modules, between $\mathbb{Z}[\beta]$ and $\mathbb{Z}[\beta^2,\beta^3]$? I can ...
8
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3answers
151 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
4
votes
2answers
79 views

$\mathbb Z_m$ where every unit is an involution

What are all $m \in \mathbb N_{\geq 2}$ such that $\forall a \in (\mathbb Z_m^*): a^2 \equiv_m 1$? Hints would be nice :) This is not homework and question 2.24 in "Introduction to Algebra" from J. ...
4
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0answers
108 views

Computing the ring of integers of a number field

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
2
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2answers
97 views

How can I write $6$ as products of irreducibles in the Gaussian Integers $\mathbb{Z}[i]$?

Moreover, how can I prove that $2+$i and $1+i$ are irreducibles?
-1
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1answer
58 views

primes of type norm($a+b\sqrt{-c}$) = primes of type $1$ $mod$ $c 2^{n-1}$?

Let $a,b,c,n$ be strictly positive integers where $c$ is a prime and such that the normed ring $a+b\sqrt{-c}$ is a UFD. I noticed the primes of norm($a+b\sqrt{-1}$) are $1 \bmod 4$. Also the ...
8
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1answer
142 views

Primes in a Power series ring

Let $\mathbb Z$ be the ring of rational integers. Consider the power series ring $\mathbb Z[[x]]$. It is known that $\mathbb Z[[x]]$ is unique factorization domain. What are the primes in $\mathbb ...
5
votes
1answer
150 views

Multiplicative Euclidean Function for an Euclidean Domain

Does there exist an Euclidean domain with no multiplicative Euclidean function? An Euclidean domain, denoted $R$, is an integral domain with an Euclidean function $d : R\setminus \{0\} \to ...
1
vote
1answer
74 views

Conjecture regarding primal representaion of vectors

I have a conjecture which I cannot prove or disprove. Denote the $i$'th digit of $x$ in binary expansion by $d_i(x)$, where for $i=1$ the MSB is taken. Example: $d_3(110.1)=0$ and $d_4(110.1)=1$ (so ...
3
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0answers
86 views

Hilbert symbol over a ring

Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows: $$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
1
vote
4answers
119 views

Perfect Square in an UFD

Let $R$ be an UFD with quotient field $F$. Show that an element $d\in R$ is a square in $R$ if and only if $d$ is a square in $F$. And then get a counterexample that above statement is not true if ...
1
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2answers
164 views

Chinese Remainder Theorem result varies

Sorry if this question is lame. First post! I was going through this book Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University In the Chapter ...
0
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1answer
70 views

Can the Euclidean algorithm prove Euclid's Lemma in a UFD?

Let $A$ be a UFD. Assume that $a,b \in A$ are relatively prime, $c \in A$ and $a | bc$. To prove that $a|c$, is the following approach correct (or do you have to use some type of prime factorization ...
0
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0answers
177 views

Struggle proving maximal ideals principal in $\mathbb Z[\varphi]$

I was worried that my proof isn't right so I want to know if there are any mistakes in this and if this way can work? Thank you very much. We want to show every maximal ideal $\mathfrak m$ of ...
1
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1answer
75 views

Infinitude of irreducibles in subring of an integer ring.

Let $\alpha \in \mathbb{C}$ be an algebraic integer of degree $n$, not a unit, and let $R = \mathbb{Z}[\alpha]$. Then every element $\beta \in R$ can be written uniquely in the form $$c_0+ c_1 ...
1
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0answers
35 views

Does an irreducible polynomial in K(t)[x] give an irreducible polynomial in K[t][x]

Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in ...
1
vote
1answer
86 views

What about the Cauchy-Frobenius-orbit-counting formula

I know the proposition that says: Let $\lambda$ be a homomorphism from a finite group $G$ into $\mathbb{C}^{\times}$. Suppose that $G$ acts on some finite set $\Omega$ and let $M$ be the number of ...
3
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3answers
136 views

Show $m^p+n^p\equiv 0 \mod p$ implies $m^p+n^p\equiv 0 \mod p^2$

Let $p$ an odd prime. Show that $m^p+n^p\equiv 0 \pmod p$ implies $m^p+n^p\equiv 0 \pmod{p^2}$.
1
vote
1answer
260 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 ...
1
vote
1answer
64 views

Algorithm to find nearest quotient in $\mathbb{Z}[i]$

Given two Gaussian integers $x$, $y$ what's the fastest way to find the Gaussian integer $z$ which minimizes $|x - zy|$? Then this Gaussian integer can be taken as $z = x/y$.
3
votes
1answer
84 views

What algorithms are there for determining whether a Gaussian integer is prime?

Give a Gaussian integer $z\in{Z[i]}$, how can I determine if $z$ is prime? I imagine there exists an algorithm that maps primality in $Z[i]$ to primality in Z. And for the case when $z\in{Z}$ I think ...
1
vote
1answer
1k views

Irreducible elements in $\mathbb{Z}[\sqrt{-2}]$ and is it a Euclidean domain?

First of all I am new to this topic, algebraic number theory, so I only know a decent (not great) amount of abstract algebra. The question I have is that, given the imaginary quadratic field ...
8
votes
1answer
267 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
25
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4answers
744 views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
1
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1answer
100 views

A property of different in Dedekind domains

Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...
1
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0answers
130 views

Easiest way to prove that a subset of even integers is closed under multiplication?

What's the easiest way of showing that; $2\mathbb{Z}\setminus (4n-2)\mathbb{Z}$ is closed under multiplication? (I'm trying to show that $(4n-2)$ is a prime element of $2\mathbb{Z}$ by showing ...
3
votes
1answer
78 views

Counting bases to which numbers are pseudoprime

Let $n=p_1^{a_1}\cdots p_k^{a_k}$ be an odd composite. Then the number of bases $1\le b\le n-1$ for which n is a strong pseudoprime is $$ \left(1 + \frac{2^{k\nu}-1}{2^k-1}\right) ...
2
votes
1answer
369 views

Norm-Euclidean rings?

For which integer $d$ is the ring $\mathbb{Z}[\sqrt{d}]$ norm-Euclidean? Here I'm referring to $\mathbb{Z}[\sqrt{d}] = \{a + b\sqrt{d} : a,b \in \mathbb{Z}\}$, not the ring of integers of ...
1
vote
1answer
79 views

Polynomials f(x) of degree at most 5 forming a ring and field

Show that the set of all polynomials f(x) of degree at most 5 with integer coefficients is a ring. Is the set of such polynomials a field? I don't see how the ring of polynomials with degree at most ...
5
votes
4answers
720 views

Finding the Units in the Ring $\mathbb{Z}[t][\sqrt{t^{2}-1}]$

This is problem taken from this link Problem 4: http://www.math.u-szeged.hu/~mmaroti/schweitzer/schweitzer-2001-eng.pdf I couldn't find the solution anywhere and i am curious to see a solution for ...
14
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3answers
948 views

Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?

I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...