All Questions
20
questions
0
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46
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Reducing an inseparable polynomial over the same field to a separable polynomial over a field
Description: Let $F$ be a perfect field and $p(x)$ a polynomial over $F$ with multiple roots. Show that there is a polynomial $q(x)$ over $F$ whose distinct roots are the same as the distinct roots of ...
0
votes
1
answer
65
views
Is $g(x)$ reducible in $k[x]$?
Let $p<q$ be primes, $k$ be a field and $f(x), g(x) \in k[x]$ be polynomials of degree $p$ and $q$ respectively.
Given :
$f(x)$ is irreducible.
$L$ is the splitting field of $f(x)$ over $k$.
$\...
1
vote
1
answer
62
views
How to obtain $\operatorname{Gal}(f\mid \mathbb{Q}_3)=A_3$ or $S_3$?
I'm doing a lot of Galois Theory lately. Now I'm more and more into $p$-adic fields. My goal is to choose an irreducible polynomial of degree $3$ over $\mathbb{Q}_2$ and $\mathbb{Q}_3$ such that you ...
- 41
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 ...
5
votes
1
answer
137
views
Some questions on polynomial $f(x)=x^4+x^2+4$ over $\mathbb{Q}$
1-) Firstly, I have shown that $f(x)$ is irreducible over $\mathbb{Q}$. To show, I first use the theorem that says: $$\text{Suppose there exist } r \in \mathbb{Q} \text{ such that } f(r)=0.\text{ Then ...
2
votes
1
answer
201
views
Splitting field of a polynomial over $\mathbb{Z}_p$
For $p$ prime, how can we show that the splitting field of $f(x)=x^{p-1}-1$ over $\mathbb{Z}_p$ coincides with $\mathbb{Z}_p$?
The roots of $f(x)$ are the $p-1$ roots of unity. I think I am correct in ...
- 99
0
votes
0
answers
46
views
Irreducibility of a polynomial over the field of rational complex functions
Let $k=\mathbb{C}(t)$ be the field of rational functions over $\mathbb{C}$.
I want to show that $P(x)=x^2+t \in k[x]$ is irreducible over $k$, and further find the degree of the splitting field of $P(...
- 99
2
votes
0
answers
72
views
Galois group of polynomial over rational functions
Let $F = \mathbb Q(y)$ be a field of rational functions, and $F[x]$ be polynomial ring over it.
How do I show that $f(x) = x^n - y$ is irreducible for a natural number $n$, and find a splitting field ...
- 81
4
votes
1
answer
2k
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Splitting field of separable polynomial is Galois extension
Definitions: $f$ is separable if every irreducible factor has distinct roots. $E/F$ is a Galois extension if the fixed field of the Galois group Gal$(E/F)$ is $F$
I would like to prove the following ...
- 1,934
0
votes
0
answers
46
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Does the degree of the splitting field of any irreducible $n$-degree polynomial equal $n$?
I am following example 50.9 of Fraleigh's first course in abstract algebra, 7 edition.
I want to find the degree of the splitting field of $f(x) = x^3 - 2$ over $\mathbb{Q}$. I verified that $f(x)$ ...
- 1,849
3
votes
0
answers
74
views
Let $p(x) = x^3 + x + 1 \in \mathbb Z_2[x]$ and $E = \mathbb Z_2[x]/p(x)$. Factor $p(x)$ into linear factors in $E[x]$. [duplicate]
Let $p(x) = x^3 + x + 1 \in \mathbb{Z}_2[x]$ and $E = \mathbb{Z}_2[x]/p(x)$. Factor $p(x)$ into linear factors in $E[x]$.
Note that $p(t) = 0$, where $t = x + ⟨p(x)⟩$. You might also wish to show ...
- 1,092
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
0
votes
2
answers
298
views
Decomposition of a biquadratic polynomial
Let $f(t):=T^4+aT^2+b$ be a polynomial in a field $K[T]$ such that it can be decomposed in two irreducible factors of degree 2.
My intuition then is screaming to me that then we neccessarily have ...
- 4,688
2
votes
1
answer
788
views
Are irreducible polynomials square free?
I have a half formed intuition that polynomials $f$ irreducible over a field $K$ have no repeated roots in its splitting field.
Is this true? What would be a proof of it?
- 4,688
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