All Questions
12
questions
2
votes
1
answer
73
views
Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?
I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
- 474
0
votes
1
answer
201
views
Splitting field of $x^3 +x +1$ over $\mathbb F_{11}$
This is a HW problem for an algebra course.
Determine the splitting field of $f(x)=x^3+x+1$ over $\mathbb F_{11}$.
I tried to use the answers from this question and this question to help me, but want ...
- 2,454
2
votes
0
answers
161
views
Splitting field of $x^8-1$ over $\mathbb{F}_2 ,\mathbb{F}_3,\mathbb{F}_{16}$
Find the splitting field of $f(x)=x^8-1$ over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_{16}$.
I tried this: We claim that the field with $q=p^m$ elements is unique. A field with $q$ elements is ...
1
vote
2
answers
319
views
Let $f ∈ \mathbb{F}_p[x]$ be an irreducible polynomial. Show that $f$ splits into linear factors in $\mathbb{F}_{p^{\deg(f)}}$.
As the title explains, I'm trying to answer the following question
Let $f ∈ \mathbb{F}_p[x]$ be an irreducible polynomial. Show that $f$ splits into linear factors in $\mathbb{F}_{p^{\deg(f)}}$.
...
- 23
3
votes
2
answers
370
views
Degree of splitting field of $x^6+tx^3+t$ in $\mathbb{F}_3(t)[x]$
I'm trying to find the degree splitting field of $x^6+tx^3+t$ in $\mathbb{F}_3(t)[x]$. After substituting $z= x^3$, and using the qudratic formula, and thensubstituting $x$ back in, I get the roots
$$...
- 1,624
0
votes
1
answer
65
views
If $f$ has $\deg(f)$ distince roots whose order are the same, then is $f$ irreducible?
P1:Let $f$ be a polynomial over finite field $F$ and $n$ be the degree of $f$. Suppose $f$ has $n$ distince roots $\alpha_i$, over its splitting field $E$, $i=1,2,\ldots,n$ and the orders of $\alpha_i$...
- 1,174
1
vote
1
answer
774
views
Does an irreducible polynomial over a finite field F divide the splitting fields polynomials for which F is a subfield?
I read somewhere that :
A subfield of $F_{p^n}$ has order $p^d$ where $d\mid n$, and there is one such subfield for each $d$.
Let $q = p^n$
We have that any irreducible polynomial of degree $n$ ...
- 153
1
vote
2
answers
128
views
Different Representations of GF(8)
Can anyone point me in the right direction with the following problem?
Given that
$$GF(8)=\frac{Z_{2}}{x^3+x^2+1}= \frac{Z_{2}}{x^3+x+1}$$
Find $\beta$ as a function of $\alpha$ , where $\alpha$ is ...
- 101
0
votes
0
answers
838
views
Show that 2 splitting fields are isomoprhic.
Let $f(x) = x^3 + 2x + 1, g(x) = x^3 + x^2 + x + 2$ be polynomials over $\mathbb{Z}_3$, let $F$ be a splitting field of $f(x)$ and $E$ be a splitting field of $g(x)$. Since $f, g$ are irreducible over ...
- 470
0
votes
2
answers
616
views
Find the splitting field of a product of two polynomials
I have trouble finding the answer to his problem and I would really appreciate an answer to this problem.
Problem : Find the splitting fields of f(x)*g(x) for f(x) = $x^3+x+1$ and g(x) = $x^3+x^2+1$ ...
1
vote
1
answer
279
views
Prove that polynomial doesn't have multiple roots over a splitting field
Let $p$ be a prime number, $m, s \in \mathbb{N}$ and $q = p^s$.
Let $f(t) = \sum_{i=0}^m \lambda_i t^{p^{i}} \in \mathbb{F}_q[t]$.
I'm asked to prove that $f(a\alpha+b\beta) = af(\alpha)+bf(\beta)$, ...
- 13.1k
1
vote
1
answer
313
views
Compute the degree of the splitting field
I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
- 453