Questions tagged [splitting-field]
The splitting field of a polynomial with coefficients in a field, $F$, is the smallest field extension to $F$ such that the polynomial decomposes into linear factors. Often used with (galois-theory).
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Defining polynomial for a compositum of splitting fields
Let $L_1,...,L_n/K$ be finite separable field extensions. Then the compositum extension $L:=L_1\cdot ...\cdot L_n/K$ is also finite and separable. Thus by the primitive element theorem, there are $\...
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How many roots are there of $(x^2-3)(x^3-3)$ in $K$, where $K$ is the splitting field of $x^3-1$ over $\mathbb F_{11}$?
Problem: How many roots are there of $(x^2-3)(x^3-3)$ in $K$, where $K$ is the splitting field of $x^3-1$ over $\mathbb F_{11}$?
I checked that only $\bar 1 \in \mathbb F_{11}$ is root of $x^3-1 \...
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Proposition 8 Corollary 1, Section 5.7 of Hungerford’s Algebra
Corollary 1.9. Let $E$ and $F$ each be extension fields of $K$ and let $u\in E$ and $v\in F$ be algebraic over $K$. Then $u$ and $v$ are roots of the same irreducible polynomial $f \in K[x]$ if and ...
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Galois group of $X^3-X+1$ over $\mathbb{Q}$ and $\mathbb{R}$ without discriminant.
Yesterday I had an exam and I had to find the galois group of the polynomial $f = x^3-x+1$. My answer was $A_3$ which is probably wrong. First of all it has no roots by the rational root theorem so it ...
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What is the Galois Group of $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?
I am studying for some algebra qualifying exams over the summer and I am stumped on the title question: What's the Galois Group for $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?
Here's what I've got so far:
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Splitting field of $x^6 + 1$ over $F_2$
So I want to find the splitting field of $g(x)=x^6+1$ over $F_2$ and the degree of the extension, so what I have done is the following
$$g(x)=x^6+1=(x^3)^2+1^2=(x^3+1)^2=(x+1)^2(x^2+x+1)^2$$
So we see ...
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Any direct method to show that $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{5}]$?
We know that Galois extension is simple extension, so the splitting field of $(X^2-2)(X^2-3)(X^2-5)$ over $\mathbb{Q}$ satisfies $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\alpha]$, for some ...
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Galois group of splitting field of $x^3-5$ over $\mathbb F _7$
Honestly, I'm not even sure where to start. I think I understand how to find the Galois group of a field extension with $\textrm{char}\mathbb F=0$ but for some reason I'm confused when it comes to ...
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Does $Z(f) \cap Z(g) = \emptyset$ implies $\gcd(f,g) = 1$? [duplicate]
If $f(x)$ and $g(x)$ are in $\mathbb{F}[x]$, where $\mathbb{F}$ is a field. Let us assume that $Z(f) = \{x \in \overline {\mathbb{F}} : f(x) = 0\}$, where $\overline{\mathbb{F}}$ is the algebraic ...
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Why does $x^3-7$ have Galois group isomorphic to $S_3$? [duplicate]
I'm not concerned with showing that the order of the Galois group is $6$; I've already done that. I'm more concerned with the structure of the Galois group. So $x^3-7$ has the roots
\begin{equation}
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Sum of nth power of some of the roots of irreducible polynomial over $ \mathbb{Q}$ is in $ \mathbb{Q}$
So i know that for a splitting field K over $ \mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $ a^n +b^n+c^n +d^n $ is in the FixGal(K,$ \mathbb{Q}$) ....
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Splitting fields and isomorphisms
If $K \subseteq L$, $K \subseteq L'$ are field extensions, $L \cong L'$ and $L$ is the splitting field of $f \in K[x]$, is $L'$ also a splitting field of $f$ ?
I think $L'$ is a splitting field $\iff$ ...
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Exercise 4, Section 5.3 of Hungerford’s Algebra
Hungerford, Algebra, page 257, gives the following as Definition 3.1:
Let $S$ be a set of polynomials of positive degree in $K[x]$. an extension field $F$ of $K$ is said to be a splitting field over $...
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Exercise 1, Section 5.3 of Hungerford’s Algebra [duplicate]
Definition: Let $S$ be a set of polynomials of positive degree in $K[x]$. An extension field $F$ of $K$ is said to be a splitting field over $K$ of the set $S$ of polynomials if every polynomial in $S$...
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Patrick Morandi "Field and Galois Theory" - Exercise I.3.12
From the book:
Let $K$ be a field, and suppose that $\sigma \in \mathrm{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K / F$ is algebraic, show that $K$ is normal over $F$.
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