All Questions
21
questions
0
votes
1
answer
93
views
Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$
How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$?
I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
- 2,636
0
votes
1
answer
303
views
Finding the $\gcd$ of two polynomials over $\Bbb Z[x]$
I understand that there are other posts on the forum about the same topic, but, after reading them, I didn't understand exactly what to do in this situation.
What I've done so far. In this same ...
- 1,141
1
vote
0
answers
55
views
Irreducible polynomials over non-UFD
It is well known that if $R$ is a UFD with field of fractions $K$ and $f \in R[x]\setminus R$, then the following holds:
$f$ is irreducible in $R[x] \iff f$ is primitive and irreducible in $K[x]$
So ...
- 349
0
votes
0
answers
93
views
Show $f(x,y)=x^7+yx^5+yx^3+3yx+y\in R[x,y]$ is irreducible in $R[x,y]$.
Let $R$ be a UFD and consider
$$
f(x,y) = x^7 + yx^5 + yx^3 + 3yx + y \in R[x,y].
$$
Show that $f(x,y)$ is irreducible in $R[x,y]$.
Proof: Note $R[x,y]=R[y][x]$. Let $S=R[y]$. We will show $f(x,y)$ is ...
- 4,653
0
votes
0
answers
83
views
Intuition behind the definition of irreducible polynomials in Gallain
The book on Abstract Algebra by Gallian defines "irreducible polynomials" as :
Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is
neither the zero polynomial nor a unit ...
2
votes
1
answer
168
views
irreducible over $\mathbb R[x,y]$
Let $A = \mathbb R[x,y]$ where $x^2 + y^2 = 1$
Show that $A$ is an integral domain.
My thoughts are to show that $x^2 + y^2 = 1$ is irreducible over $\mathbb R[x,y] = \mathbb R[x][y],$ I am guessing ...
user889267
11
votes
1
answer
183
views
Is a polynomial $y^n+y^{n-1}-x^m-x^{m-1}$ irreducible in $\Bbb Z[x,y]$?
The question is in the title, $n>m\ge 2$ are integers. All text below is the context.
Two weeks ago user759001 asked on integer solutions $x>y\ge 2$ of a Diophantine equation
$$x^{m-1}(x+1)=y^{n-...
- 93.3k
2
votes
0
answers
49
views
Why a polynomial is irreducible over Z
Prove that the following polynomial is irreducible over $\mathbb{Z}$:
$$f(x) = x^8-x^7+x^5-x^4+x^3-x+1$$
My attempt: one can see that $f(x)=(x^4+x^3+1)(x^4+x+1) $ over $\mathbb{F_2}$ where these 2 ...
user708427
5
votes
1
answer
145
views
Show that $1-999x^{888}\in \mathbb{Q}[x]$ is irreducible
As the title says, I'm supposed to show $1-999x^{888}\in \mathbb{Q}[x]$ is irreducible.
In a previous part of the question I had to show $x^{888} -999\in \mathbb{Q}[x]$ was irreducible which I did ...
- 53
0
votes
0
answers
277
views
A primitive polynomial of positive degree is irreducible over D[X] iff it is irreducible over $F[X]$
I'm trying to prove this result from Hungerford's Algebra (Lemma 6.13):
Let $D$ be a unique factorization domain with quotient field $F$ and $f$ a primitive polynomial of positive degree in $D[X]$. ...
- 456
3
votes
1
answer
107
views
Factorization in $\mathbb{Z}[x]$?
In my rings and modules course we've learned some results relating factorizability of polynomials in $\mathbb{Z}[x]$ to factorizability of polynomials in $\mathbb{Q}[x]$ and $\mathbb{F}_p[x]$. Some of ...
- 11.5k
0
votes
0
answers
242
views
A consequence of Gauss' lemma
The following is written in Fulton's Algebraic curves book but I can't quite understand it.
If $R$ is a UFD with quotient field $K$, then (by Gauss) any
irreducible element $F\in R[X]$ remains ...
- 3,493
0
votes
2
answers
125
views
Prove that $2x^4+15x^2+10$ is irreducible in $Q[x]$
The problem suggests using a suitable change of variable of the form $y=ax+b$ and using Eisenstein's criterion to show that it's reducible in $\mathbb{Q}[x]$. So, I have several doubts, first of all, ...
1
vote
0
answers
296
views
Polynomial of degree $5$ reducible over $\mathbb Q(\sqrt 2)$
Give an example of an irreducible monic polynomials of degree (a) $4$; (b) $5$ in $\mathbb Z[x]$ that is reducible over $\mathbb Q(\sqrt 2)$, or prove that none exists.
I managed to find a polynomial ...
- 12k
-1
votes
1
answer
2k
views
Every irreducible Polynomial is primitive or irreducible constant [closed]
Let $R$ be an UFD. Is it true that an irreducible polynomial $f \in R[T]$ is either:
primitive
$f \in R$ and irreducible in $R$
?
- 1,275