All Questions
Tagged with prime-factorization abstract-algebra
97
questions
0
votes
1
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184
views
factor $29$ into irreducible elements in $\mathbb{Z}[i\sqrt{ 2}]$ and $\mathbb{Z}[ \sqrt {7}]$
The statement goes as the title.
So I figured that $29 = (5+2i)(5-2i)$ in $\mathbb{Z}[i]$. However I'm not sure how to continue since the respective rings look like this:
$\mathbb{Z}[i\sqrt{ 2}]=\{a+...
- 77
0
votes
0
answers
233
views
Prime elements in the ring of integers adjoin a complex number.
Let $R=\mathbb Z[e^{i\pi/3}]=\{a+be^{i\pi/3}\mid a,b\in \mathbb Z\}\subseteq \mathbb C$
(a) Show that $R$ is a Euclidean domain using the Euclidean norm $N(u)=|u|^2$.
(b) Show that if $p$ is a prime ...
- 149
1
vote
1
answer
33
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Why is it that the degree of a subextension of $K(a^{1/p})/K$ must have a degree dividing $p$?
$p$ is a prime
$K$ is of characteristic not $p$
$a∈K$
$a^{1/p}∉K$
$a$ is not a root of unity
"Then if we pick an element $b∈K(a^{1/p})$, $b∉K$, then $K(b)/K$ is a non-trivial subextension, thus of ...
- 5,166
1
vote
2
answers
136
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Justifying the representation of a monic polynomial over a UFD
Let $D$ be a UFD and $f(x) \in D[x]$ be monic. The book I'm reading from claims that $$f(x) = p_1(x)^{e_1} \cdots p_n(x)^{e_n}$$
where $p_i(x)$ are distinct, irreducible, and monic, and $e_i >0$.
...
- 4,039
3
votes
1
answer
688
views
Primary Decomposition
The standard primary decomposition theorem in algebra is about being able to write an ideal uniquely as an intersection of primary ideals. In linear algebra the theorem is about how a vector space can ...
- 1,457
2
votes
1
answer
162
views
Factorization in the Sense of Primes [closed]
Are there types prime factorizations in the sense of primes other than the Fundamental Theorem of Arithmetic (and its generalization to euclidean/principle rings), primary decomposition and Dedekind ...
- 1,457
2
votes
1
answer
306
views
A simple algebraic ring extension of a UFD having no prime elements
Let $D$ be a UFD over a field $k$ of characteristic zero.
Assume that $w$ is algebraic over $D$.
Denote $R=D[w]$.
Observe that $R$ is not necessarily a UFD.
Can one find an example in which $R$ ...
- 6,705
3
votes
1
answer
84
views
A non-UFD $B$ such that $A \subset B \subset C$, where $A \cong C$ are UFD's
Let $A \subset C$ be two isomorphic unique factorization domains (UFD's).
Is it possible to find an integral domain $B$, $A \subset B \subset C$,
such that $B$ is not a UFD?
I have tried to ...
- 6,705
0
votes
1
answer
73
views
Prime power decomposition
$x^{147} \equiv (((x^{7})^{7})^{3})\equiv x^{3}(mod7)$
How does $x^{147}$ simplify into $x^{3}(mod7)$
What Corollary is responsible for this?
Edit:
Fermat's Little Theorem is needed:
147 = 3 * 7 *...
- 289
2
votes
1
answer
744
views
There exist a prime factorization for all algebraic irrational numbers?
I was wondering:
If $a, b \in \mathbb{N}$ and obviously they both have a prime factorization, we know that $a+b \in \mathbb{N}$ and $a+b$ also has a prime factorization so $a$, $b$ and $a+b$ can be ...
- 2,597
0
votes
1
answer
359
views
Can the linear polynomials be totally ordered?
At the start of this document http://www.math.lsa.umich.edu/~lagarias/575chomework/p-adic-chap5.pdf Lagarias draws the analogy between writing any integer in base $p$, and writing any polynomial in ...
- 9,299
1
vote
3
answers
114
views
$a\times na = LCM(a, b) = b\times nb$. Show that $GCD(na, nb) = 1$.
I have this question that I have been trying to get my head around for the past couple of hours and I am not getting anywhere with it...
Let $a,b \in \mathbb{N}$. Further, let $n_a, n_b \in \mathbb{...
- 361
2
votes
1
answer
71
views
Proof involving modular and primes
My Question Reads:
If $a, b$ are integers such that $a \equiv b \pmod p$ for every positive prime $p$, prove that $a = b$.
I started by stating $a, b \in \mathbb Z$.
From there I have said without ...
- 1,088
3
votes
0
answers
180
views
Does every commutative ring with $1 \neq 0$ and a non-zero, non-unit element contain a prime element?
As above, I am trying to answer the following question:
"Does every commutative ring with $1 \neq 0$ and a non-zero, non-unit element contain a prime element?"
Since in some books prime elements are ...
- 4,899
0
votes
2
answers
76
views
Proof using Prime Decomposition [duplicate]
My question reads:
If c^2=ab and (a,b)=1, prove that a and b are perfect squares.
I began my proof by $a=p_1 p_2 \cdots p_n$ and $b=q_1 q_2\cdots q_m$. Then I gave $c$ its own decomposition as ...
- 1,088