Questions tagged [extension-field]
Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.
5,007
questions
1
vote
0
answers
14
views
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 $\...
- 113
1
vote
1
answer
81
views
Let $b=\sqrt[3] 2$, $c=-\frac{1}{2}+\frac{\sqrt {-3}}{2}$ with $a=bc$. Prove $b+a$ and $b-a$ has degree $3$ and $6$ over $\mathbb Q$, respectively.
Let $b=\sqrt[3] 2$, $c=-\frac{1}{2}+\frac{\sqrt {-3}}{2}$ with $a=bc$ in $\mathbb C$. Prove that $b+a$ and $b-a$ has degree $3$ and $6$ over $\mathbb Q$, respectively.
My attempts: $b+a=b+bc=b(1+c)=\...
- 1,150
1
vote
0
answers
36
views
Inclusion of field extensions $\mathbb{Q}(\omega_p)$ and $\mathbb{Q}(\omega_{2p})$
Could you help me clear up a confusion? Either my following reasoning is wrong, or I should be able to show that $\omega_{2p} \in \mathbb{Q}(\omega_p)$, which does not seem correct.
Degree of Field ...
- 131
1
vote
1
answer
50
views
Natural way to extend the ring $\mathbb{Z} / p^k \mathbb{Z}$ so that the equation $x^2 + 1 \equiv 0 (\text{mod }p^k)$ has a solution
We know that for $p \equiv 3 (\text{mod }4)$, there is no solution to $x^2 + 1 \equiv 0 (\text{mod }p^k)$ for $k = 1, 2, \ldots$, by quadratic reciprocity. But can I embed the ring $\mathbb{Z} / p^k \...
- 556
1
vote
1
answer
61
views
Inertia field example of $ \mathbb{Q}_5(\sqrt[4]{50})$
Let $L = \mathbb{Q}_5(\sqrt[4]{50})$ and denote by $E$ the inertia field of the extension $L / \mathbb{Q}_5$. Write down a prime element $\pi_E $ of $ \mathcal{O}_E $ with $L = E(\sqrt{\pi_E})$. Can ...
0
votes
1
answer
25
views
extension field in the subring $K[u]$
If $F$ is an extension field of a field $K$, $u, u_i \in F$, and $X \subset F$, then
\begin{enumerate}[(i)]
\item the subring $K[u]$ consists of all elements of the form $f(u)$, where $f$ is a ...
- 1
0
votes
1
answer
30
views
On Frobenius–Schur indicator of real/complex representations
Let $G$ be a finite group with complex irreps $W_i$. Let $V$ be a real irrep of $G$. Denote $\chi_{W_i}$ and $\chi_{V}$ the corresponding characters.
Each $V$ has three possibilities:
Case 1: $\dim_{\...
- 2,307
1
vote
1
answer
67
views
Is there a separable extension of degree 21?
Given the field $F= \mathbb{F}_3$ and a transcendental $t$, I am trying to find an intermediate field
$$ F(t^{1/63}) \supset E \supset F(t) $$
where $[F(t^{1/63}): E]= 21$ and the extension is ...
- 3,044
0
votes
1
answer
27
views
Determining the Equality of Two Field Extensions
Let $F$ be a field of characteristic $0$. Let $F(\alpha)/F$ be a finite extension of degree not divisible by $3$. Is is true that $F(\alpha^3)=F(\alpha)$? If we assume that they are not equal, since $\...
- 730
2
votes
1
answer
35
views
Easier way to show finite simple extensions have only finitely many intermediate fields
I was reading this post that concerns the idea that simple finite field extensions have only finitely many intermediate fields between them. I'm wondering why the following method hasn't been proposed:...
- 2,780
0
votes
2
answers
47
views
Purely inseparable field extensions - proving $\alpha^{p^m} \in F$ implies $m_\alpha = x^{p^m}-a^{p^m}$
I'm reading Isaacs' "Algebra: A Graduate Course" and I don't really understand the proof for the implication (2) $\Rightarrow$ (3) in Theorem 19.10 (page 298):
Suppose $F\subseteq E $ is an ...
- 485
1
vote
0
answers
68
views
Are "other" algebraic extensions of $\mathbb{R}$ similar to $\mathbb{C}$? [duplicate]
If we define $\mathbb{R}[x] = \{ c_0 + c_1 x + \cdots + c_n x^n : n\in\mathbb{N}, \forall c_k \in \mathbb{R} \}$ as the set of polynomials with coefficients in $\mathbb{R}$ then I'm familiar with the ...
- 4,305
1
vote
1
answer
66
views
Finding the fixed field corresponding to Galois group $\textrm {Gal} (\mathbb Q (\root p \of 2, \zeta_p)$ for prime $p$.
I've already shown that this is the splitting field of the polynomial $x^p-2$ over $\mathbb Q$, and that the degree of extension of $L=\mathbb Q (\root p \of 2,\zeta_p)$ is $p(p-1)$. Let $H$ be the ...
- 485
0
votes
1
answer
47
views
Possibly inseparable extensions of $\mathbb{F}_p((t))$
I have a question on local fields. Some sources define a characteristic $p$ local field as of the form $k((t))$ where $k/\mathbb{F}_p$ is a finite extension. Some sources define it as a finite ...
- 725
3
votes
4
answers
162
views
Determine if $\mathbb Q (\root4 \of 5i,\sqrt3)$ is a normal extension of $\mathbb Q$.
I've shown that $\mathbb Q (\root4 \of 5i,\sqrt3)$ is of degree 8, and that the roots of $x^4-5$ are $\root4 \of 5i,-\root4 \of 5i,\root4 \of 5,-\root4 \of 5$. I'm having trouble checking whether $\...
- 485