All Questions
Tagged with finite-rings number-theory
14 questions
1
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0
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42
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Uniqueness of common $ \mathbb Z _ n $-roots of polynomials in $ \mathbb Z [ x ] $ from linearity of their GCD
$ \def \Z {\mathbb Z} $EDIT: In the statement of the problem, the "every $ n \in \Z _ + $" is now changed to "all but finitely many $ n \in \Z _ + $". The unnecessary previous ...
- 6,896
1
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1
answer
178
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Linear congruences over a finite ring are equivalent
Let $n,N$ be two natural numbers. Let $a=(a_1, ..., a_n),b=(b_1, ..., b_n)\in (\mathbb{Z}/N\mathbb{Z})^n$ with $\gcd(a_1, ..., a_n, N) =1$ and $\gcd(b_1, ..., b_n, N) =1$. Define $H_a = \{(x_1,...,x_n)...
user1154495
4
votes
1
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149
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A polynomial with unique root in every $ \mathbb Z _ n $
Let $ p ( x ) \in \mathbb N [ x ] $ be a polynomial with nonnegative integer coefficients, and $ a \in \mathbb Z $ be a given integer constant. If for all positive integers $ n $, $ p ( x ) + a $ has ...
- 6,896
2
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1
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696
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Classification of quadratic forms over $\mathbb{Z}/n\mathbb{Z}$ - even characteristic case
Let $R$ be a ring (unital, commutative) and $M$ a free $R$-module of finite rank. A quadratic form is a map $q:M\rightarrow R$ such that
$\forall r\in R:\forall m\in M: q(rm)=r^2\cdot q(m)$ and
the ...
- 2,024
3
votes
1
answer
137
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Finite quotients of ring of integers of local field
Let $K$ be a non-Archimedean local field, so either a finite extension of $\mathbb{Q}_p$ or a finite extension of $\mathbb{F}_q((t))$. Let $\mathcal{O}$ denote its ring of integers and $\pi$ a ...
- 3,085
4
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2
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53
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Stuck: Finding an Isomorphism for an Invertible Ring
I'm stuck on a problem creating an isomorphism between rings. Specifically, let $\mathbb{Z}[\sqrt{7}] = R$.
Then for the invertible group $(R/3R)^\times$, I want to find an isomorphism to another ...
5
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138
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Let $R$ be a finitely generated subring of a number field. Is $R/I$ finite for every non-zero ideal of $R$?
Given any finitely generated subring $R$ of a number field (finite extension of $\mathbb{Q}$) or a global function field (finite extension of $\mathbb{F}_p(T)$), does $R$ have the property that $R/I$ ...
- 3,970
0
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0
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135
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Zero-Sum Partitions of Nonzero Elements of a Ring
In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
- 50k
2
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1
answer
604
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the number of zero divisors in polynomial ring
I was looking for an answer on the question
How much zero divisors are in the ring $\dfrac{\mathbb{Z}_3[x]}{(x^4 + 2)}$?
when I came up with the brilliant/hack-isch idea that it might just be $81 -...
1
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0
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173
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How many unique combinations of sets can we get?
Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists.
Set $p$ a prime greater than 2, $\alpha = (p-1)/2$, and $\beta=\...
- 6,179
0
votes
1
answer
96
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Is this "sliding window" unique?
Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists.
Set $p$ a prime greater than 2, $\alpha = \lfloor p/2 \rfloor$, ...
- 6,179
5
votes
1
answer
321
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Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?
I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly ...
- 21.1k
1
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2
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186
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Simple Combinatorics in finite rings
Let $g = [g_{1} g_{2} \dots g_{r}] \in \Bbb Z_{q}^{*r}$ be a given vector with each $g_{i} \in \Bbb Z_{q}^{*}$ where $\Bbb Z_{q}^{*}$ is $\Bbb Z_{q} \backslash \{0\}$ and $q > 6$ is odd. How many ...
- 6,273
11
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2
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3k
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Solving systems of linear equations over a finite ring
I want to solve equations like this (mod $2^n$):
$$\begin{array}{rcrcrcr} 3x&+&4y&+&13z&=&3&\pmod{16} \\ x&+&5y&+&3z&=&5&\pmod{16} \\ 4x&+&...
- 1,088