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.

Filter by
Sorted by
Tagged with
2 votes
0 answers
29 views

$E$ generated as a $K$-vectorspace by the coefficients of quotient of minimal polynomials.

Question: Let $K \subset E \subset L = K(\alpha)$ be a tower of field extensions, with $\alpha$ algebraic over $K$. Prove that $E$ as a $K$-vectorspace is generated by the coefficients of the ...
ByteBlitzer's user avatar
0 votes
0 answers
26 views

Is the degree of the extension the same as the size of the Galois group of the polynomial?

Suppose that $f \in K[X]$ is a separable irreducible polynomial. Let $L$ be the splitting field of $f$ over $K$. Is it true that $[L:K] = |Gal(f|K)|$? Why I think this is because, first of all $|Aut(L|...
blomp's user avatar
  • 35
0 votes
1 answer
33 views

trouble identifying number of elements in small Fields adjoin polynomial solns

If $G(x)=x^2+x+1$ and $H(x)=x^3-x+1$ where $r_G$ and $r_H$ are roots of $G(x)$ and $H(x)$ respectively, what do the elements of (I). $\mathbb{F}_2(r_G)$, (II). $\mathbb{F}_2(r_H)$, (III). $\mathbb{F}...
ness's user avatar
  • 358
0 votes
0 answers
47 views

Is $\mathbb{Q}(\sqrt(2)\sqrt[3](5) \subset \mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$?

Is $\mathbb{Q}(\sqrt(2)\sqrt[3](5)) \subset \mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$? I've tried writing any element $x \in \mathbb{Q}(\sqrt(2)\sqrt[3](5)$ with $x = a + b\sqrt(2)\sqrt[3](5)$ in terms of $a +...
Mathh's user avatar
  • 1
0 votes
0 answers
48 views

over a Field extension

Let $f(x) = ax^2+bx+c \in \Bbb{R}[x]$ be an irreducible polynomial. Prove that the field $\Bbb{R}[x] \mathbin{/} \langle f(x) \rangle$ is isomorphic to the field of complex numbers. Knowing that $$ \...
Marco's user avatar
  • 47
0 votes
1 answer
50 views

Automorphism group isomorphic to the Klein 4 group.

Question: Prove that $\text{Aut}(\mathbb{Q}(i, \sqrt{2})) \cong V_{4}$, where $V_{4}$ is the Klein 4 group. Attempt: We know that under an automorphism $\sigma \in \text{Aut}(\mathbb{Q}(i, \sqrt{2}))$ ...
ByteBlitzer's user avatar
0 votes
1 answer
55 views

Does $F(\alpha) = F + (\alpha)$?

Let $\alpha \in K$ for $F$ is a field and $K$ is an extension of $F$. Then $F(\alpha)$ is a simple extension of $F$ i.e. $F(\alpha)$ is the smallest subfield containing $F$ and $\alpha$. Is it ...
Grigor Hakobyan's user avatar
0 votes
0 answers
18 views

Show: the splitting field of $f = a·p_1^{m_1}\cdots p_t^{m_t}$ is identical to that of the polynomial $p_1\cdots p_t$

I wanted to check my solutions for this problem: Let $f ∈ K[T]$ with prime factorization $f = a·p_1^{m_1}\cdots p_t^{m_t}$ , with $a ∈ K^{×}$ and pairwise different normalized and irreducible ...
Marco Di Giacomo's user avatar
1 vote
1 answer
52 views

$K = \mathbb C(X)[T]/(X^2-f(T))$ is purely transcendental when $f$ has a double root

Let $f \in \mathbb C[T]$ be a monic polynomial of degree $3$, with $z_1,z_2,z_3$ as roots. Denote $K=\mathbb C(X)[T]/(X^2 - f(T))$. I have to prove that if $z_1=z_2$, $K$ is purely transcendental and ...
mathcounterexamples.net's user avatar
-2 votes
1 answer
89 views

How $\alpha$ is a root of $x^2-\alpha^2$ in $F[\alpha^2][x]$? [closed]

I was reading the following solution here Prove that $E=F[\alpha^2]$ and I was not sure how is this statement in a solution is correct? "since $\alpha$ is a root of $x^2-\alpha^2$ in$F[\alpha^2][...
Emptymind's user avatar
  • 1,869
0 votes
0 answers
42 views

Extending the Fundamental Theorem of Projective Geometry to projective lines over quadratic field extensions

Is there a theorem which characterises which bijective maps $f:\mathbb{GP}^1 \to \mathbb{GP}^1$, where $\mathbb{G:F}$ is a field extension of degree 2, have the property that whenever the cross ratio ...
wlad's user avatar
  • 8,175
1 vote
1 answer
30 views

Show degree of minimal polynomial has to be even

We define the polynomial $f(T) := T^6 - 3T^2 + 1 \in \mathbb Q[T].$ Let $\alpha \in \mathbb C$ be a root of $f.$ I showed that $f \mod 3$ can be factorized into three irreducible factors of degree $2:$...
Minerva's user avatar
  • 153
1 vote
1 answer
45 views

$L/K$ field extension and $[M : K] ≤ C$ holds for all real intermediate fields $M$. Then $[L : K]$ is also finite

I want to check my solution for this exercise: Let $L/K$ be a field extension and there exists a constant $C ∈ \mathbb{N}$ such that $[M : K] ≤ C$ holds for all proper intermediate fields $M$. Then $[...
Marco Di Giacomo's user avatar
1 vote
0 answers
29 views

Every ring $R$ with $K ⊆ R ⊆ L$ (subrings) is a field $\Leftrightarrow$ $L/K$ is algebraic

I have this exercise where I am having some problems solving it: Show that for a field extension $L/K$ the following are equivalent: (a) $L/K$ is algebraic. (b) Every ring $R$ with $K ⊆ R ⊆ L$ (...
Marco Di Giacomo's user avatar
3 votes
2 answers
139 views

Is there an algorithm to check if a polynomial with integer coefficients is reducible over some extension of $\mathbb Q$?

I have been reading some solved questions here, for example: Showing a polynomial is irreducible over an extension field. Is $f(y) = 5 - 4y^2 + y^4$ irreducible over $\mathbb{Q}(\sqrt{5})$? showing ...
NotaChoice's user avatar
1 vote
1 answer
46 views

Show $\mathrm{disc}(1,\alpha,\dots,\alpha^{d-1})=(-1)^{d(d-1)/2}N_{\mathbb Q(\alpha)/\mathbb Q}(f'(\alpha))$, with $d=[\mathbb Q(\alpha):\mathbb Q]$

Suppose $K = \mathbb{Q}(\alpha)$ for some $\alpha$ algebraic, and let $f$ be the minimal polynomial of $\alpha$ over $\mathbb{Q}$. Say $d = [K:\mathbb{Q}]$. I want to show that $$\mathrm{disc}(1, \...
Robin's user avatar
  • 3,185
0 votes
1 answer
26 views

the degree of the discriminant of a cubic irreducibble polynomial over the rational adjoint a root

Let $f(x) \in \mathbb Q[x]$ and $$f(x)= x^3 + px +q=(x-r_1)(x-r_2)(x-r_3),$$ and $r_1, r_2,r_3\in K$ for the splitting field $K$ of $f(x)$. Let the $D = (r_3 - r_1)^2(r_2- r_1)^2(r_3 - r_2)^2$, which ...
Iris's user avatar
  • 17
2 votes
2 answers
74 views

Factorization of ideals in number fields of the form $K=\mathbb{Q}(\alpha,\beta)$

Given a number field $K=\mathbb{Q}(\alpha)$, I am able to factorize ideals of the form $p\mathcal{O}_K$ where $p$ is some unramified prime, via the Dedekind Kummer theorem. Now I have two questions: ...
Ninja's user avatar
  • 2,777
0 votes
1 answer
63 views

Finite extension of finite field

I have two questions: Suppose $F$ is a finite field then prove that it is a simple extension of its prime subfield. Suppose $F$ is a finite extension of a finite field $K$. Then prove that $F$ is a ...
RIPAN DAS's user avatar
1 vote
0 answers
28 views

"Galois group" acts transitively on irreducible factors in the context of normal extensions (not necessarily separable)

Let $k$ be a field and $K$ be a finite normal extension of $k$. Let $G$ be the group of automorphisms of $K$ over $k$ (since $K/k$ is not necessarily separable, this group may not be called Galois ...
Degenerate D's user avatar
2 votes
1 answer
87 views

Regarding the isomorphism $PGL(2,K) \cong GL(2,K)/Scal(2, K)$ and a degree of a field extension

I was originally trying to show this isomorphism: $$Aut_K(K(x)) \cong PGL(2, K) \cong GL(2, K)/Scal(2, K), \text{ where } K \text{ is a field, and } Scal(2,K) \text{ is the group of scalar 2x2 ...
rikusp2002's user avatar
2 votes
0 answers
31 views

$\bar{x}$ notation in Field Theory [duplicate]

in my lecture about field extension and tower theorem for extension degree, the professor gives the following example : $$\mathbb{Z}_2 \le \frac{\mathbb{Z}_2[x]}{(x^2 + x + 1)} \le \frac{(\frac{\...
EtiBeranger's user avatar
0 votes
1 answer
39 views

Linear equations in field extension

Let $F\subset E$ be a finite field extension of degree $2$ and $\,$ $^{-}:E\rightarrow E$ the conjugacy map. Consider the equation of the form $$ ax+b\overline{x}=c, \qquad a,b,c\in E. $$ Under what ...
QMath's user avatar
  • 399
0 votes
1 answer
63 views

Irreducibility of $f$, automorphisms $\sigma \in \text{Aut}_{\mathbb Q}(L)$

Let $f \in \mathbb Q[X]$ and $f(X) = (X - \alpha_1)\cdots (X - \alpha_n)$ with $\alpha_1, \dots, \alpha_n \in \mathbb C$ pairwise distinct. Let $L = \mathbb Q(\alpha_1, \dots, \alpha_n)$ be the ...
Minerva's user avatar
  • 153
3 votes
0 answers
52 views

Characterizing all primitive elements of a simple extension

Let $p, q \in \mathbb{Q}[T]$ be irreducible and with degree greater than one (and WLOG monic). When is it that $\mathbb{Q}[T]/(p(T)) \cong \mathbb{Q}[T]/(q(T))$? Take any roots $\alpha, \beta$ of $p, ...
Arthur Queiroz Moura's user avatar
1 vote
3 answers
122 views

A commutative diagram for $F(\alpha) \cong F[x] / \langle p(x) \rangle$

I've been trying to understand the following proposition: Let $E$ be a field extension of $F$ and $\alpha \in E$ be algebraic over $F$. Then, $F(\alpha) \cong F[x]/\langle p(x) \rangle$ where $p(x)$ ...
pedropedro's user avatar
0 votes
0 answers
67 views

Questions of The Extension of Field [duplicate]

Let $E_i (i=1, 2)$ be subfields of $K$ containing the subfield $F$, and $[E_i : F] < \infty$. If $E$ is the subfield of $K$ generated by $E_1$ and $E_2$, then $[E:F] \leqslant [E_1 : F][E_2 : F]$. ...
Klein's user avatar
  • 1
1 vote
0 answers
63 views

Lagrange resolvent in field extensions

Here is a question I encountered in school: Show that $\mathbb{Q}\left(\zeta_{21}\right)$ has exactly three subfields of degree 6 over $\mathbb{Q}$. Show that one of them is $\mathbb{Q}\left(\zeta_7\...
Arthur M.'s user avatar
1 vote
0 answers
49 views

What is known of this problem similar to the inverse Galois problem? Galois groups of $\mathbb{Q}(X)/\mathbb{Q}(h(X))$ for $h \in \mathbb{Q}(X)$

What is known about the answer to the following question? What finite groups can be realised as the Galois group of finite Galois extensions of the form: $$\mathbb{Q}(X)/\mathbb{Q}(h(X))$$ for $h(X)$ ...
Robin's user avatar
  • 3,185
1 vote
2 answers
169 views

Proving $\sqrt[3]{5} \notin \mathbb{Q}(\sqrt[3]{2})$ with solvability of system in $\mathbb{Q}$

We were asked to show that $\sqrt[3]{5} \notin \mathbb{Q}(\sqrt[3]{2})$ by reducing the problem to the solvability of system in $\mathbb{Q}$. There was also a hint with it that a system does not need ...
rikdb's user avatar
  • 85
1 vote
2 answers
72 views

Relations between field extensions of $\mathbb{Q}$

We were the following related questions, with $\omega$ as the third root of unity: Determine the relation between the following field extension: $\mathbb{Q}(\sqrt{3}\omega)$, $\mathbb{Q}(\sqrt{3}+\...
rikdb's user avatar
  • 85
1 vote
0 answers
30 views

For $K$ a field of characteristic $p$, $[K(X):K(X^p-X)]=p$.

I want to show for $K$ a field of characteristic $p$, that $[K(X):K(X^p-X)]=p$. Say $X^p-X=Y$. I want to argue something along the lines of $Z^p-Y-a \in K(Y)[Z]$ is irreducible, if $a$ is a root, then ...
Robin's user avatar
  • 3,185
0 votes
0 answers
35 views

Understanding proof of $\text{Tr}_{M/K}(\alpha) = \text{Tr}_{L/K}(\text{Tr}_{M/L}(\alpha)), \, \forall \alpha \in M$

I had asked previously today about what the author of this proof meant by $\text{Tr}_K(\varphi : V \to V)$ in the statement of lemma 9.20.4. Now I'm trying to understand the proof itself, so I created ...
Arthur Queiroz Moura's user avatar
0 votes
1 answer
69 views

Help understanding proof of $\text{Tr}_{M/K}(\alpha) = \text{Tr}_{L/K}(\text{Tr}_{M/L}(\alpha)), \, \forall \alpha \in M$

I'm trying to understand this proof (lemma 9.20.5, which is based on lemma 9.20.4). But I don't understand the proof of lemma 9.20.4. It involves $\text{Tr}_K(\varphi : V \to V)$. The definition for $\...
Arthur Queiroz Moura's user avatar
10 votes
3 answers
3k views

Why can't we extend any field by simply adding a new symbol to it?

After trying to recall some fundamental field theory, I got very confused at the notion of field extensions. For example, when we make $\mathbb{C}$ out of $\mathbb{R}$, we can simply think of it as ...
Jesus's user avatar
  • 1,788
-1 votes
2 answers
100 views

Field homomorphism $\phi: F \to F$ with $F$ extension over $\mathbb{Q}$ is identity map

While reviewing exercises for introduction to algebra final I came across the following question: Consider a field extension $F$ over $\mathbb{Q}.$ Now suppose we have a field homomorphism $\phi: F\to ...
rikdb's user avatar
  • 85
0 votes
0 answers
75 views

$K[\alpha]=K(\alpha)$ (if and) only if $\alpha$ is algebraic? [duplicate]

I think I have totally confused myself and just wanted to check whether the only if in the statement above even holds. Let us look at some field extension $K/L$ and some $\alpha\in L$. We proved in ...
dancingqueen's user avatar
1 vote
1 answer
80 views

Show that $A[T]/(P(T))$ is a local flat A-algebra where $(A,\mathfrak{m},k)$ is a local ring with $k[T]/(\tilde{P}(T))$ is a simple extension of $k$

Let $(A,\mathfrak{m},k)$ be a (Noetherian) local ring. Let $k[T]/(\tilde{P}(T))$ be a simple field extension of $k$ with $\tilde{P}(T)$ monic and irreducible (and separable). Lift $\tilde{P}(T)$ to ...
Z Wu's user avatar
  • 1,691
0 votes
0 answers
37 views

Degree of algebraically closed field extensions

Let $\ell$ be a field extension of the field $k$, and let $[\ell : k] = n$ (possibly infinite). Let $\overline{\ell}$ and $\overline{k}$ be algebraic closures of $\ell$ and $k$. Is it true that $[\...
Boccherini's user avatar
2 votes
1 answer
58 views

If $E/F$ and $K/E$ are simple extensions, then $K/F$ is simple?

If $K/F$ is a finite extension and $E$ is an intermediate field, then I can see that $K/F$ being simple implies $K/E$ and $E/F$ are simple since $K/F$ has a finite number of intermediate fields. ...
PNB's user avatar
  • 33
2 votes
1 answer
76 views

Degree of the perfection of a field

I am currently studying perfection in the context of Galois Theory. For a field $K$ of characteristic $p$ and algebraic closure $K'$, we define $$ K^{\text{perf}} = \{ a \in K' : \text{there exists } ...
rulerandcompass's user avatar
0 votes
0 answers
25 views

Separable degree of an intermediate extension in a normal extension

I am currently studying the separable degree of an algebraic extension. I have come across the following result: if $K \subset L \subset M$ are algebraic extensions and $M$ is normal over $K$, show ...
rulerandcompass's user avatar
-3 votes
1 answer
60 views

Find the minimal polynomial for a root over the rationals

Find the minimal polynomial over $Q$ of the root $(\sqrt[2]{3})/(1+\sqrt[3]{2})$. I'm new to field theory so please help! (I'm new to the platform so excuse my sloppy notation) So far, I set a= $(\...
PND's user avatar
  • 3
1 vote
0 answers
126 views

Degree of $F(\alpha, \beta)$ over $F(\alpha + \beta)$

Let $F$ be a field. I have a separable, irreducible polynomial $f(x) \in F[x]$ and $\alpha$ and $\beta$ are two distinct roots of $f$. I want to show that $[F(\alpha, \beta): F(\alpha + \beta)] \geq 2$...
Dalop's user avatar
  • 663
-4 votes
1 answer
107 views

Proof that {1, pi} generates the extension R/Q [closed]

I was trying to find a counterexample for: If E/K is a finitely generated extension then [E:K] must be finite I know there is a proposition that states: E/K is a finitely generated extension and ...
Cram245's user avatar
0 votes
1 answer
54 views

Action of some Galois group on scheme of finite type

Let $X$ be a $k$ variety (i.e. a $k-$scheme of finite type). Let $k^{s} \subset \bar{k}$ be the separable closure of $k$. I will wright $X(k^{s})$ for the set of $k$ morphism from $Spec(k^{s})$ to $X$....
Analyse300's user avatar
1 vote
0 answers
59 views

Construct primitive element for intermediate field of cyclotomic field extension

I am studying subfields of cyclotomic extension, especially the following situation: Let $\xi$ be a primitive $m$-th root of unity, and $\chi$ be a multiplicative character on $(\mathbb{Z}/m\mathbb{Z})...
isz's user avatar
  • 31
0 votes
0 answers
24 views

Degree of extension of $\mathbb Q(\sqrt{a_1},\cdots,\sqrt{a_n})$ over $\mathbb Q$ [duplicate]

Assume $a_1,\cdots,a_n \in \mathbb N_+$, none of which is perfect square and each pair of them is coprime. Proof that $$ [\mathbb Q(\sqrt{a_1},\cdots,\sqrt{a_n}): \mathbb Q]=2^n $$ I have tried a lot ...
QiFeng233's user avatar
0 votes
1 answer
83 views

Valuation on $K(x, y)$ similar to $\mathfrak{p}$-adic valuation

Let $K$ be a field. Since $K[x]$ is a Dedekind domain, constructing valuations for $K(x)$ is easy - we can just take any prime ideal of $K[x]$ and consider the $\mathfrak{p}$-adic valuation. So ...
Gauss's user avatar
  • 2,461
0 votes
1 answer
54 views

If $E/F$ is a Galois extension, $Gal(E/F)$ is finite, and $K$ is an intermediate field that is a degree 2 extension from $F$, prove $K/F$ is Galois

I am a bit stuck on how I am meant to approach this. Firstly, I think of the fact that, as $E/F$ is Galois, $|Gal(E/F)|=|E:F|=|E:K||K:F|=|E:K|2$. Then, I know that an intermediate field of a Galois ...
cable's user avatar
  • 126

1
2 3 4 5
98