Often called prime polynomials. Polynomials that have no polynomial divisors.

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5
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4answers
147 views

Trivial question about splitting fields

I'm having trouble with super basic ring theoretic concepts. Given some irreducible $f$ in $k[x]$, denote by $p$ the projection to the quotient $F$ by the ideal generated by $f$. I'm struggling to ...
0
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2answers
47 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
0
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1answer
26 views

How can I show that $(x+1)$ and $(x^2+x+1)$ are irreducible in $\mathbb{R}$?

How can I show that $(x+1)$ and $(x^2+x+1)$ are irreducible in $\mathbb{R}$? For $(x+1)$, I'm not sure if it suffices to say $(x+1)$ has degree 1 so it is irreducible?
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1answer
44 views

Checking whether a given polynomial is reducible or irreducible.

We're given the polynomial $x^{2}-2$ , and we need to prove that it's irreducible in $\mathbb Q$ but reducible in $\mathbb R$. Writing the polynomial as $(x^{2}-2) = 1.(x^{2}-2)$ , $(x^{2}-2)$ ...
2
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2answers
60 views

Show $p(x)$ is a primitive polynomial

First the definition: Polynomial $q(x) \in \mathbb{Z}_p[x]$ of degree $n$ is called primitive, iff: $q(x) \mid x^{p^n-1}-1$ $\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$ ...
1
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1answer
39 views

Is $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$?

I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in ...
0
votes
1answer
10 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
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0answers
17 views

irreducible polynomials over $\mathbb{F}_p[x_1,x_2]$

Recently I was reading about a irreducibility test for polynomials over the ring $\mathbb{F}_q[x]$. It is layed on the fact that the product of all monic irreducible polynomials whose degree divides ...
0
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2answers
37 views

On analogy between $\Bbb Z$ and $\Bbb F_q[x]$

There are objects and operations analogous between $\Bbb Z$ and $\Bbb F_q[x]$. For example primes in $\Bbb Z$ and irreducibles in $\Bbb F_q[x]$ are analogous and so is multiplication operation. ...
0
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3answers
78 views

List all elements of $\mathbb{Z}_5[x]/\langle x^2+3x+1\rangle$

List all elements of $\mathbb{Z}_5[x]/\langle x^2+3x+1\rangle$ Is there a simple way to solve these kind of questions (only using pen and paper)? If I understand it correctly, I can multiply ...
0
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1answer
22 views

Simplifying Rational Expressions in a Finite Field Extension

In Dummit and Foote's textbook one of the exercises is: Let $\theta$ be a root of $x^3-2x-2$ over $\mathbb{Q}$. Compute $\frac{1+\theta}{1+\theta+\theta^2}$ in $\mathbb{Q}(\theta)$. My approach ...
0
votes
1answer
41 views

Proof of equality of $2$ polynomials without using Galois Theory?

I have the following situation and ask myself whether an argument is available to prove the equality of the $2$ following polynomials. Both polynomials are of degree $n$ and are monic. The ...
0
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0answers
25 views

Is there an “easier” way to find the factors of a polynomial than using Ruffini's method?

I am a first year Mathematics student, and sometimes I have a hard time with Ruffini's method for polynomials, specially in the field of Rationals
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1answer
39 views

Multiplicative Inverse in a $256$ Galois Field

I am working on finding the multiplicative reverse in $GF(2^8)$ using the Euclidean Algorithm but after reading multiple sources, I feel as though I am proceeding incorrectly. Using the irreducible ...
0
votes
0answers
16 views

How to separate a real integer polynomial into two with real/complex roots?

If the coefficients of a polynomial p(x) are all real integers, then every root of p(x) is either 1) real or 2) a complex number whose conjugate is also a root of p(x). Is there any easy way to ...
1
vote
1answer
40 views

Prove $x^n+1$ is irreducible over $\mathbb{Q}[X]$ iff $n=2^k$ for $k \in \mathbb{N}$

Unfortunately, I cannot find any information on or anything similar to this particular question. Might be quite new. In all honesty, I don't know how to tackle either side of this question. By ...
5
votes
3answers
49 views

State whether $x^5-5x^4+10x^3-7x^2+8x-4$ is irreducible or not

I have tried everything in my knowledge and no, I cannot state it. I have tried a factorizor online which tells me that it is not factorizable hence irreducible. But I cannot reason why. I looked at ...
1
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0answers
37 views

Irreducible polynomials and Eisenstein's criterion.

I am looking for a counter-example for the following: If $p(x)$ is an irreducible polynomial over $Z[x]$, then there is a polynomial in $Z[x], q(x)$, so that $p(q(x))$ is irreducible by Eisenstein's ...
1
vote
1answer
42 views

Why is $x^3 + 3x + 2$ irreducible by plugging in elements of $\mathbb{Z}_5$

The polynomial $x^3+3x+2$ is irreducible in $\mathbb{Z}_5[x]$. I get that it must take the form of $(x-a)g(x)$ where $a$ is a zero,but my book plugged in elements of $\mathbb{Z}_5$ to show no zeros, ...
1
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2answers
50 views

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$. I tried use the Eisenstein Criterion (with a change variable) but I have not succeeded. Thanks for your help.
2
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1answer
24 views

What is meant by “maximal proper factors” of a integer?

I understand what is meant by proper factors, e.g. the proper factors of 36 are 2, 3, 4, 6, 9, 12, & 18. However I've just seen the phrase "maximal proper factors" used in the context of ...
2
votes
1answer
29 views

Stuck with the statement: $t^4+2$ in $\mathbb{Z}_5$ gives rise to…

This is from Ian Stewart's book on Galois theory, I am looking at irreducible polynomials. It talks of irreducibility over mod. It takes as an example, $f(t)=t^4+15t^3+7$ over integers, and asks us ...
2
votes
1answer
62 views

How can I get a irreducible polynomial of degree 8 over $Z_2[X]$? [closed]

I have got one of degree 5: $x^5+x^2+1$, but I need one of degree 8.
1
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1answer
35 views

Difference between irreducible factors in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$

I don't fully understand the difference between factorizing a polynomial in irreducible factors in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. For example $f=X^4-X^2+4X+3$ is irreducible in $\mathbb{Z}[X]$, ...
1
vote
1answer
41 views

Factorizing $X^4-Y^2$ in $\mathbb{Q}[X,Y]$

I want to factorize $X^4-Y^2$ in $\mathbb{Q}[X,Y]$ in irreducible factors. I thought about using Eisenstein's criterium to show that it is irreducible, though I'm not sure what the prime element is ...
0
votes
1answer
35 views

Irreducible polynomials in $\mathbb{C}[X,Y]$

I have the polynomial $X^2+Y^2-1$ in $\mathbb{C}[X,Y]$. Is this irreducible? If not, how do I factorize it? Should I handle this the same as if it were $\mathbb{R}[X,Y]$, or should I do it ...
5
votes
1answer
155 views

Showing a polynomial irreducible

How to show that the polynomial $Y^2+X^2(X-1)^2$ is irreducible in $\mathbb R[X,Y]$. I tried to show that $\mathbb R[X,Y]$ modulo this ideal is an integral domain but I cannot find any homomorphism.
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2answers
63 views

Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$

I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$. The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to ...
0
votes
1answer
9 views

Eisenstein criterion on almost cyclotomic polynomial

Let's consider the equation: $f(X) = x^5 + x^4 + x^3 + x^2 + x + 1$ How does one proof with the Eisenstein criterion (it has to be with this one) that this polynomial is irreducible in ...
2
votes
1answer
35 views

How to factor a polynomial of degree 4 that is the product of two irreducible quadratic polynomials

It is easy to construct a polynomial of degree four with integer coefficients that doesn't have any real roots, if it has a real root then it can be factored by division by $x-a$ where $a$ is the root ...
0
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2answers
38 views

Show that sum of roots is rational.

If $f(X)$ in $\mathbb{Q}[X]$ is an irreducible polynomial polynomial of degree $n \geq 2,$ with roots $\alpha_1, \alpha_2,\ldots,\alpha_n$ in $\mathbb{C},$ show that $\displaystyle ...
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2answers
32 views

Irreducibility of a Given Polynomial

I'm starting to work with polynomial rings, and I've gotten to some problems related to irreducibility. I am trying to see if I'm approaching this problem correctly and how I can move forward with a ...
4
votes
3answers
53 views

Monic irreducible polynomials of degree 6 in $F_{5}[X]$

Question A How many monic irreducible polynomials of degree 6 in $F_{5}[X]$ Question B Give an example of an irreducible polynomial of degree 6 in $F_{5}[X]$ Idea for a Such a polynomial would be ...
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5answers
97 views

Proving $x^4+2$ cannot be factored into $2$ degree polynomials

My book says that it can't be because if I try to write $x⁴+10x³+15x²+5x+12$ as: $$x^4+2$$ (which is $p(x) mod 5$) then $x⁴+2$ is irreducible because: $$x^4+2 = (ax²+bx+c)(a'x²+b'x+c)$$ is ...
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0answers
9 views

$d(x)$ is an gcd $iff$ $a\cdot d(x)$ is a gcd ($a\neq 0)$

My definition for gcd is that: If $d(x)$ is a common divisor of $p_1, \cdots, p_n$, and for any other $d'(x)$ that is also a common divisor, this $d'(x)$ will divide $d(x)$, then $d(x)$ is the ...
4
votes
3answers
56 views

Find all irreducible polynomials of degree $2$ over $\mathbb{Z}_5$

Obviously if I write all the possible ones and try the roots I'd get a LOT of polynomials $(125)$ and I'd have to test $5$ roots for each of them, which would be a LOT. Is there any idea? I must also ...
0
votes
4answers
87 views

Irreducibility of $x^4-5$ over $\mathbb{Z}_{17}$

Obviously, it'd be hard to try all the $17$ elements to see if there is some root, and even if there is none, it'd be necessary to verify if it can't be factored into two irreducible 2 degree ...
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2answers
29 views

Which of the sets are ideals and maximal ideals?

The exercise asks me to prove which of the sets are ideals, and if they are, which of those are maximal. I have these 4 cases: $$ a) J = \{f(x)\in \mathbb{Q}[x]: f(1)=f(7)=0 \} \\b) J = \{f(x)\in ...
0
votes
2answers
49 views

Proving some polynomials are irreducible using Eisenstein's criterion

I would like to see if I'm right about these polynomials I tried to prove are irreducible: 1) For the first polynomial I used that if a polynomial is irrational over $\mathbb{Z}_p$ for $p$ prime, and ...
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1answer
52 views

Geometrically ireducible curve

I know that curve with coefficients in $k$ is geometrically ireducible if it does not factor over algebraic closure of $k$. I have this curve, for example, $$2x^2+2x^2y+2y^2+2xy+3xy^2=1.$$ It's ...
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0answers
16 views

$p$ a nonsingular point in $V$ and $V(f)\cap T_p(V)$ then $f|_{T_p(V)}$ has a factor of multiplicity $\geq 2$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. at. Let $V \subset k^n$ be a hypersurface with $I(V ) = \langle f \rangle$. Show that if $V$ is not a hyperplane and $p \in V$ ...
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0answers
10 views

regarding primitive polynomials

Definition given in the book: Let $R$ be a factorial ring. It is said a polynomial $f\in R[X]$ is primitive if its coefficients are pairwise without common divisor. It's also said that from the given ...
2
votes
3answers
101 views

If $f(x,y)\in R[x,y]$ is an irreducible polynomial, is $R[x,y]/f(x,y)$ a field?

If $f(x,y)\in R[x,y]$ is an irreducible polynomial, is $R[x,y]/f(x,y)$ a field? I know that this is true for $R[x]$, because of the Euclidean algorithm. However, is it also true for polynomial ...
1
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0answers
25 views

Which of the following field properties are correct?

Let $\omega = \cos{\frac{2\pi}{10}}+i\sin{\frac{2\pi}{10}}$. Let $K = \mathbb{Q}(\omega^2)$ and $L = \mathbb{Q}(\omega)$. Then $[L : \mathbb{Q}] = 10.$ $[L : K] = 2$. $[K : ...
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0answers
26 views

Show that $X^p-X-1\in \mathbb Q[X]$ is irreducible.

How can I show that $X^p-X-1\in \mathbb Q[X]$ is irreducible ? I tried reduction modulo $p$, but doesn't work, even some normal variable changing as $X\longmapsto X+1$ or $X\longmapsto X-1$, but ...
3
votes
2answers
39 views

Check if $X^3 -2$ is irreducible or not in $\mathbb{Z}_{31} [X]$

Check if $f(X):= X^3 -2$ is irreducible or not in $\mathbb{Z}_{31} [X]$ I can note that if $f(X)$ is reducible I have one factor of degree $1$ and so there is at least one root of $2$ in ...
1
vote
1answer
49 views

Decomposing polynomials over $\mathbb{Z}_3$

The first polynomial I had to decompose over $\mathbb{Z}_3$ is: $$x^2+x+1$$ I started by noticing that one root of it is $1$ so I thougth that I could factor this polynomial by $(x-1)$ but I ...
10
votes
5answers
211 views

How to factor $x^6+x^5-3\,x^4+2\,x^3+3\,x^2+x-1$ by hand?

I know that $x^6+x^5-3\,x^4+2\,x^3+3\,x^2+x-1 = (x^4-x^3+x+1)(x^2+2x-1)$ but I would not know how to do that factoring without a software. Some idea? Thank you!
1
vote
1answer
52 views

Decomposing $x^4-5x^2+6$ over some fields

My book asks me to decompose $$x^4-5x^2+6$$ over: $K = \mathbb{Q},\\ K = \mathbb{Q[\sqrt{2}]},\\ K = \mathbb{R}$ For $K = \mathbb{Q}$, I substituted $x² = a$ to get: $$a²-5a+6 = (a-3)(a-2)$$ So ...
1
vote
3answers
28 views

Showing that $1$ is the only root in $f=x^5 -1 \in \mathbb{F}_p[x]$ if $p-1$ is not divisible by $5$

I am trying to show that if $p-1$ is not divisible by $5$ then $1$ is the only root in $f=x^5 -1 \in \mathbb{F}_p[x]$, where $p$ is a prime. I can see that $f = (x-1)(x^4 + x^3 + x^2 + x + 1)$ so it ...