Tagged Questions
4
votes
3answers
45 views
Factorize in R[x]
I have the polynomial $x^8+1$, I know that there's no root for solve this in $\Bbb R[x]$ but i want to factorize this to the minimal expression. This is possible or this is irreducible?
2
votes
0answers
31 views
Find the factorization of the polynomial as a product of irreducible [duplicate]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the ...
1
vote
1answer
36 views
Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$
Testing with the simplest possible root in this case, $P(1) = 0$
...
6
votes
1answer
47 views
Calculating in quotient ring of $\mathbb{R}[X]$
Part of an old Oxford exam (1992 A1)
We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square.
Now, we note first that ...
4
votes
2answers
93 views
Irreducibility of $x^n-x-1$ over $\mathbb Q$
I want to prove that
$p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible.
My attempt.
GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
4
votes
1answer
43 views
What's the difference of naming a polynomial ring as $\mathbb{C}\{ x,y\}$ and $\mathbb{C} [x,y]$?
I sometimes see both notations and I am led (maybe misled) to believe that they are the same thing. What is the formal difference between both of them? Or there isn't any?
5
votes
1answer
79 views
General proof that a product of nonzero homogeneous polynomials is nonzero (under certain conditions).
Background, Notation, Definitions: Given a set $X$, I define the set $M(X)$ of monomials with $X$-indeterminates to be the set of elements of $\omega^X$ having finite support. Given $m_0,m_1\in M(X)$, ...
4
votes
2answers
44 views
Idempotents in a Quotient Ring
Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$.
I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
6
votes
3answers
74 views
Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies
Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime.
Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
1
vote
0answers
33 views
Equality of two $k$-algebras
Let $f\in k[X_1,\ldots, X_n]$ and $1-fX_{n+1}\in k[X_1,\ldots, X_{n+1}]$. Moreover $X\subseteq k^n$ is a subset and
$$I(X)=\{g\in k[X_1,\ldots, X_n]\,:\, g(x)=0\,\forall x\in X \}$$
is the ideal of ...
0
votes
1answer
22 views
How to calculate the quotient and the reminder when $F=\mathbb Z_5[x],f(x)=3x+1,g(x)=x^3+2x+1?$
Division Algorithm says that, for any field $F$ and for $f(x),g(x)(\neq 0)\in F[x]$$~\exists$ unique $q(x),r(x)\in F[x]$ such that $f(x)=g(x)q(x)+r(x)$ where $\deg r(x)<\deg g(x)$ or $r(x)=0.$
I'm ...
0
votes
0answers
77 views
How to show an ideal is zero-dimensional? [duplicate]
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional.
How do I go about showing this?
2
votes
1answer
65 views
Prove that $D[x]$ is an integral domain if $D$ is one.
Prove if $D$ is an integral domain and $f,g\in D[X]$ are nonzero, then $fg$ does not equal $0$ and $\deg[f(x)g(x)]=\deg f(x) + \deg g(x)$.
I do not know much about this since I just learned about it. ...
1
vote
2answers
62 views
What is “prime factorisation” of polynomials?
I have the following question:
Find the prime factorisation in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreduciblity in $\mathbb{Z}[x]$, of ...
2
votes
1answer
49 views
Subring of Z[x] generated by set of integers and polynomials
Let Z be the ring of integers. We have the subring of Z[x] generated by integers and p1 and p2 (p1 and p2 are polynomials over Z, we note it as Z[p1,p2]). I've got for my homework to investigate if ...
0
votes
2answers
23 views
In $Z_5 [x]$, the monic GCD of the polynomials $(x+[4])(x+[3])$ and $([3]x + [2])(x + [3])$ is $(x + [3])$
True or False
In $Z_5 [x]$, the monic GCD of the polynomials $(x+[4])(x+[3])$ and $([3]x + [2])(x + [3])$ is $(x + [3])$.
my solution :
$([3]x+[2])$ is $[3](x+[4])$ therefore gcd is ...
6
votes
1answer
85 views
Vandermonde identity in a ring
Let $R$ be a commutative $\mathbb{Q}$-algebra. For $r \in R$ and $n \in \mathbb{N}$ we can define the binomial coefficient $\binom{r}{n}$ as usual by $\binom{r}{0}=1$ and ...
2
votes
2answers
89 views
$\mathbb{C}[X,Y]/(Y-X^2)\cong\mathbb{C}[X]$
Let $K$ be a field and $f(X)\in K[X]$. Then we have a well-defined
surjective homomorphism
$$\varphi: K[X,Y]/(Y-f(X))\to K[X]$$
given by $[g(X,Y)]\mapsto g(X,f(X)$.
Someone has ...
1
vote
2answers
40 views
What are maximal ideals of $K[t]$?
Let $K[t]$ be the algebra of all polynomials in $t$. What are maximal ideals of $K[t]$? I know that $\langle t \rangle = \{tf \mid f \in K[t]\}$ is a maximal ideal. Are there other maximal ideals? ...
2
votes
1answer
97 views
Reduced Gröbner Basis for $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$
I have $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$ with lex $X>Y>Z $.
I have calculated the Gröbner Basis as $G=\{ X^2+2XYZ, XY+2Y^2Z-1, X, 2Y^2Z-1 \}$.
But the question I have asks for the Reduced ...
8
votes
3answers
146 views
Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$
We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows:
Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
2
votes
2answers
73 views
Algebraic Independence of Equations vs Polynomials
I am considering the difference between algebraic independence of a system of equations and polynomials. Are these two notions equivalent? For example, for $x, y, z$ real,
$xy = A$
$yz = B$
$xz = ...
3
votes
1answer
122 views
Showing that a ring homomorphism is injective
Let $f\in\mathbb{Z}[X]$ be a monic irreducible polynomial, $\alpha$ a root of $f$ and $k\in \mathbb{Z}$. Show that the map
$$\varphi : \mathbb{Z}/f(k)\mathbb{Z} \to \mathbb{Z}[\alpha]/(k - ...
2
votes
3answers
71 views
Show that the ideal of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$is a prime ideal.
Show that the ideal $I$ of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$is a prime ideal, where $k$ is a field.
I tried to prove it by contradiction. Suppose $f$ and $g$ are not of ...
3
votes
8answers
184 views
Showing two polynomial rings over $\mathbb{C}$ aren't isomorphic
Im trying to show that the ring of polynomials in one variable over the complex numbers is not isomorphic to the ring over $\mathbb C$ with two variables $x$ and $y$ modulo $\langle x^2-y^3\rangle$. ...
8
votes
2answers
82 views
$R$ with an upper bound for degrees of irreducibles in $R[x]$
One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as
If $f\in\mathbb{R}[x]$ has degree larger ...
1
vote
1answer
93 views
How many solutions to $X^6-1=0$ in $\mathbb{Z}/(504)$
How many roots of $X^6-1$ are there in $\mathbb{Z}/(504)$
I believe it's easily resolvable with the abelian groups fundamental theorem, but I want a solutions which uses only the basic notions of ...
4
votes
0answers
77 views
Some elementary facts
What is the simplest and the most conceptual proof of some basic facts on algebraic geometry?
1) Hilbert's Nullstellensatz
2) Regular functions on projective variety - only constants
3) elemination ...
5
votes
1answer
97 views
Perron polynomial irreducibility criterion
Facts before question:
$\textbf{Fact 1:}$ Let $F(X) = X^n + a_{n-1}X^{n-1} + \cdots + a_1X+a_0\in \mathbb{Z}[X]$, with $a_0\neq 0$.
If $|a_{n-1}|>1+|a_{n-2}| + \cdots +|a_1| + |a_0|$, then $F$ ...
0
votes
3answers
207 views
$R = \mathbb{Z}[ i ] / (5)$ is not an integral domain? Why?
Let $R = \mathbb{Z}[ i ] / (5)$ .
How should I prove that $5 = (2+i) (2-i)$ is a prime factorization in $\mathbb{Z}[i]$?
Can we deduce from this that R is not an integral domain? How?
I know that ...
0
votes
1answer
46 views
Example of non-unique-factorization in $\mathbb{Z}+x \mathbb{Q}[x]$.
Could you provide me an example of a polynomial that is not uniquely factorized in $\mathbb{Z}+x \mathbb{Q}[x]$? Thanks!
1
vote
1answer
310 views
How does one prove that a polynomial has no rational roots in general?
How can we prove that a polynomial only has rational roots when we know the coefficients and the degree? For instance, in illustration, how would we show this for $x^8 ...
1
vote
2answers
226 views
Factoring polynomials of degree 6 in 2 ways.
Let $P(x)$ be an integer polynomial of degree $6$ that is irreducible over the integers.
$P(x) = x^6 + (A+a) x^5 + (B+ aA+ b) x^4 + (C+aB+bA +c) x^3 + (aC +bB +cA) x^2 + (bC+cB) x + cC = x^6 + ...
0
votes
2answers
163 views
Ideals in Polynomial Rings
$I=\langle x^2,2x,4\rangle$ is an ideal of $\Bbb Z[x]$.
Prove that $I$ is not a principal ideal and find the size of $\Bbb Z[x]/I$.
Using the theorem that ideals are principal iff the generator is ...
3
votes
2answers
140 views
Factoring in Z3[x]
I need to factor $x^6+x^4+x^2+1$ into irreducible parts in $Z_3[x]$. Obviously this polynomial reduces to $(x^4+1)(x^2+1)$ which is irreducible in $Z[x]$, but I'm not sure how to confirm that it's ...
3
votes
2answers
82 views
$F[x]/\langle f(x) \rangle$ has $q^n$ elements
today I have a problem which I see in book Abstract Algebra of David. This is problem:
Let $F$ be a finite field of order q and let $f(x)$ be a polynomial in $F(x)$ of degree $n\geq 1$. Prove that ...
2
votes
1answer
66 views
Polynomials Question
Let $R$ be a commutative ring with unity, $a\in R$, $f(x) \in R[x]$. Then $a$ is a zero of $f$ iff $x-a$ is a factor of $f$.
Solution: If $a$ is a zero of $f$, then by division algorithm, we can ...
-1
votes
2answers
246 views
$R[x]$ has a subring isomorphic to $R$.
Say $R$ is a commutative ring. Does there exists a subring of $R[x]$ that is isomorphic to $R$?
My approach would be to define the subring of $R[x]$ that generates $R$. Any thoughts?
0
votes
1answer
104 views
Problem related polynomial ring over finite field of intergers
if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.
0
votes
0answers
86 views
$\mathbb{Z}[x]/\langle x,2 \rangle$ is prime and maximal ideal
Let $\mathbb{Z}[x]/\langle x,2 \rangle$, We know $\langle x,2 \rangle = \{f \in \mathbb{Z}[x] : f(0)$ is even integer$\}$
Also $$\mathbb{Z}[x]/\langle x,2 \rangle = \{f(x) + \langle x,2 \rangle\} = ...
0
votes
2answers
65 views
Relation between torsion subgroup of multiplicative group of field and solvability of polynomials
In a broad sense, what relationships are there between the torsion subgroup $G$ of the multiplicative group of non-zero elements of a field $K$ and whether or not certain polynomials in $K[x]$ have ...
1
vote
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 ...
0
votes
1answer
36 views
Is $ U = \{f(x)| f(x) \in P_{3}, \operatorname{deg} f(x) = 3\}$ a subspace of $P_{3}$?
Given : $ U = \{f(x)| f(x) \in P_{3}, \operatorname{deg} f(x) = 3\}$
Does: U is a subspaces of $P_{3}$
I think the answer is yes. But in my textbook, they say no. And explain that zero is not in ...
0
votes
3answers
65 views
Conjugation of polynomial in $\mathbb{Z}[x]$.
Let $a,b \in \mathbb{Q}$ and $d \neq 0,1$ be a square free integer.
Define $\overline{a + b\sqrt{d}} = a - b\sqrt{d}$
If $f \in \mathbb{Z}[x]$ show that: $f(\overline{\alpha}) = ...
0
votes
2answers
42 views
Irreducibility of $x^{3}-t\in\mathbb{C}(t)[x]$
Denote $F=\mathbb{C}(t)$ and consider $p(x)=x^{3}-t\in F[x]$
Is it true that $p$ is irreducible over $F$ ?
My thoughts:
I think that since it is not true that $t^{2}\mid t$ (I don't know
how to ...
1
vote
2answers
65 views
The Uniqueness of a Coset of $R[x]/\langle f\rangle$ where $f$ is a Polynomial of Degree $d$ in $R[x]$
Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$. How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$? ...
0
votes
0answers
94 views
Factoring polynomials $f(g(x))$ over extension fields.
This question is a variation on another one :
related question
Let $f(x)$ and $g(x)$ be polynomials with integer coefficients and discriminant of $f(x)$ > discriminant of $g(x)$ , which are ...
5
votes
3answers
77 views
$(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$?
I'm trying to see why the equation $(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$ holds in the power series ring $\mathbb{Z}[[t]]$. I assume it's a counting argument about the number of ...
1
vote
1answer
99 views
Factoring polynomials of degree $a p^b$ over extension fields.
Let $f(x)$ be an irreducible polynomial with integer coefficients, which is irreducible over $\mathbb{Z}$ and has degree $a p^b > p$ with $p,a,b>0$ and $p$ a prime.
It appears that $f(x)$ ...
5
votes
1answer
117 views
polynomials over a local Artinian (or finite) ring
In this question " Zero-divisors and units in $\mathbb Z_4[x]$ " it looks like it has been shown that the set of zero divisors of $\mathbb{Z}_4[x]$ coincides with its nilpotent elements.
Since the ...
