Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

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4
votes
1answer
139 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
0
votes
1answer
24 views

Product field of intermediate fields is field extension

Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension. 1 How do I prove that $EL$ is a finite ...
1
vote
2answers
60 views

$\operatorname{char}R=0 \implies\mathbb{Q} \hookrightarrow R$

Let $R$ be any field, then: $$\operatorname{char}R=0 \implies \mathbb{Q} \hookrightarrow R$$ Proof: We know that $\mathbb{Q} = Q(\mathbb{Z})=\{[(x,y)]\subseteq\mathbb{Z}\times \mathbb{Z^*}:(x,y) ...
0
votes
2answers
51 views

Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
0
votes
2answers
36 views

Splitting field of $x^5-3x^3+x^2-3$

I am trying to solve the following problem, Find the degree of the splitting field of the polynomial $p(x)=x^5-3x^3+x^2-3$ over $\mathbb{Q}.$ My approach for solution: Clearly -1 is a root of the ...
2
votes
2answers
44 views

If Q(a,b) is a field extension, can we always choose an equivalent extension Q(c) such that c=a+b?

If we have two complex numbers $a,b$ that are algebraic over $\mathbb {Q} $, we can make an extension $\mathbb {Q}(a,b)$ that is equal to an extension $\mathbb {Q}(c)$ for some $c\in \mathbb {C} $. ...
3
votes
0answers
31 views

Infinite extensions of “finite degree under $\mathbb{Q}$” [duplicate]

Consider an algebraic extension $K$ of $\mathbb{Q}$. The degree $[K:\mathbb{Q}]$ of $K$ is defined as the dimension of the extension considered as a vector space. Now, let $\overline{\mathbb{Q}}$ be ...
0
votes
0answers
26 views

Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
0
votes
0answers
18 views

Frobenius Map and Subfields of $\bar{\mathbb{F}}(x,y)$

Let $L=\bar{\mathbb{F}}_{p}(x,y)$ (for $\bar{\mathbb{F}_{p}}$ an algebraic closure of the finite field) and let $L^{(p)}$ be the image of $L$ under the Frobenius map $L \rightarrow L $, where $g(x) ...
1
vote
1answer
41 views

Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$

Find the Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$, for $\zeta_{3}$ being a third primitive root of unity. It's easy to show this is a Galois extension since it will be ...
1
vote
1answer
41 views

$\mathbb{Z}_p \hookrightarrow R$ is it necessary $R$ commutative?

Let $R$ be an integral domain with $\operatorname{Char}(R)=p$, with $p$ prime. Then: $$\mathbb{Z}_p \hookrightarrow R$$ The proof is not difficult. My questions are: 1) Is it necessary to have an ...
-1
votes
1answer
45 views

$X^4-10X+1$ reducible in $\mathbb{F}_p[X]$ for all prime $p$ [duplicate]

Show that the polynomial $X^4-10X+1$ is irreducible in $\mathbb{Z}[X]$ but reducible in $\mathbb{F}_p[X]$ for all prime $p$. I could show the irreducibility in $\mathbb{Z}[X]$ but not sure how to ...
2
votes
0answers
40 views

char$(K)=0$ then $K(x^{2}) \cap K(x^{2}-x)=K$ [duplicate]

Let $K$ be a field of characteristic zero and $K(x)$ the field of rational functions with coefficients in $K$. Let $K(u)$ denote the subfield of $K(x)$ generated by $u \in K(x)$ over $K$. My ...
3
votes
2answers
64 views

Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
6
votes
1answer
166 views

Is there a (not so) generalized version of Hilbert's Theorem 90?

I'm sorry if my following question doesn't make any sense. We know that if $L/k$ is a finite Galois extension then $H^{1}(\mathrm{Gal}(L/k),L^{*})=0$ (Hilbert's theorem 90). However I would like to ...
0
votes
2answers
25 views

In a field of characteristic 0, for any integer $m$ and an element $x$, does there exist another element $y$ that $ym=x$?

As the title. Or rather, for any integer $m$ which is not the characteristic, does such an 'integer division' exist?
0
votes
0answers
40 views

Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field? I know for a fact that the general form is true, ...
0
votes
0answers
15 views

A normal closure of an arbitrary field extension

Let $L/K$ be an arbitrary algebraic field extension. How is a normal closure of $L$ (the smallest normal extension of $K$ containing $L$) constructed? If $L/K$ is finite, then writing ...
0
votes
1answer
21 views

Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?

Let $F$ be an extension field over $K$ such that $F(x)$ is a finite extension over $K(x)$ ; then is it true that $[F:K]$ is also finite ? ( I know about the converse , that if $F/K$ is a finite ...
0
votes
1answer
75 views

If $x^p−x−c$ is irreducible in $F[x]$ then it has no root in the field.

The complete problem appears in Hungerford's Algebra. Let $c\in F$, where $F$ is a field of characteristic $p$ ($p$ prime). Then $x^p−x−c$ is irreducible in $F[x]$ if and only if $x^p−x−c$ has no ...
0
votes
1answer
16 views

Proving an element belongs to field extension

I am unsure of questions asking to prove that an element belongs to a field extension. Here is an example: Prove that $\sqrt2 \in \mathbb{Q}( \sqrt{11+3\sqrt{13} } )$ $\sqrt2 \notin ...
2
votes
2answers
39 views

Normal closure of $\mathbb{Q}(\sqrt{11+3\sqrt{13}})$ over $\mathbb{Q}$

The following is a question from an undergrad course in Galois theory: Find a normal closure $L$ of $K=\mathbb{Q}(\sqrt{11+3\sqrt{13}})$ over $\mathbb{Q}$ I know that normal extensions are ...
3
votes
1answer
31 views

If $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ and $[K : \mathbb{Q}]=p$ then $K= \mathbb{Q}(\sqrt[p]{2})$

Let $p,q$ be distinct prime numbers and assume that $p<q$. Prove that if a subfield $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ satisfies $[K : \mathbb{Q}]=p$, then $K= ...
0
votes
0answers
17 views

Normal transcendental extension

According to Wikipedia, normal extension are assumed to be algebraic. But one of the definitions $K/k$ is normal if any $k$-embedding $\sigma : K \rightarrow \Omega$ of $K$ into a fixed algebraic ...
2
votes
1answer
43 views

Example of finite field extension where root not separable

My class notes has as theorem (without proof): "Let $K/F$ be finite field extension, with $K=F(\alpha_1,\ldots,\alpha_n)$ and $\alpha_k$ is separable for all $k$. Then $K/F$ is separable". My ...
0
votes
1answer
30 views

If every polynomial in $F[x]$ splits then there exists no nontrivial algebraic extension

Im trying to prove the statement of the title: If every polynomial in $F[x]$ splits then $F$ has no nontrivial algebraic extension I was thinking about arguing as follows: if there existed an ...
0
votes
0answers
30 views

Complex Norms when D = 1 mod 4

Let $D ∈ \mathbb Z$ and let $\alpha ∈ \mathbb C$ be such that $\alpha^2 = D$. Let $\beta = \frac{1+\alpha}{2}$ and $\overline{\beta} = \frac{1-\alpha}{2}$ if $D = 1$ mod $4$ and $\beta = \alpha$, ...
1
vote
1answer
52 views

There is no field with exactly 6 elements

I saw the related posts, and I tried a different proof. Please have a look. Let $D$ be any field with $|D|=6$. $|D|=6<\infty \Longrightarrow Char(D)\neq 0\Longrightarrow Char(D)=prime\ number$ ...
0
votes
0answers
26 views

Find the degree of a tower of field extensions

Let $E = F(\alpha, \beta)$ be an extension of the field $F$. We're given that the minimal polynomial of $\alpha$ in $F[x]$ is of degree $d_1$, and the minimal polynomial of $\beta$ in $F[x]$ is of ...
0
votes
0answers
16 views

$ \forall a\in U(R) : ord(a)=Char(R) $

Theorem: Let $(R,+,\cdot)$ be a ring with unity $1_R$. Then $$ \forall a\in U(R) : ord(a)=Char(R) $$ Proof: If $ord(a)=n$, $ord(1_R)=m=Char(R)$ then $n1_R=n(a \cdot a^{-1})=(na) \cdot a^{-1}=0_R ...
0
votes
2answers
21 views

Suppose $\gamma$ is the root of some irred. polynomial in F[x], why is [F($\gamma$):F($\gamma^3$)] $\leq$ 3

I have verified the inequality for a concrete case, but I'm not sure how to show that it is generally true. How can this be proven? Also, if we replace 3 by some other number, will analogous ...
1
vote
1answer
33 views

splitting field of $x^n-1$ over $\mathbb{Q}$

Is it true that the splitting field for $x^n-1$ over $\mathbb{Q}$ is $\mathbb{Q}(\xi_n)$ where $\xi_n$ is a primitive n$^{th}$ root of unity, making it an extension of degree $\phi(n)$ (Euler phi ...
0
votes
1answer
37 views

Stuck on last part of rings $\mathbb{Z}[x]/(x^2 + 7)$ and $\mathbb{Z}[x]/(2x^2 + 7)$ isomorphic?

I am checking to see if the rings $\mathbb{Z}[x]/(x^2 + 7)$ and $\mathbb{Z}[x]/(2x^2 + 7)$ isomorphic? I want to assume that the two rings are isomorphic and let $f$ be the isomorphism. I can let A = ...
1
vote
1answer
28 views

Splitting field for $x^4-x^2-2$

Am i right to say that the splitting field for $x^4-x^2-2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2},i)$ which is of degree 4? i.e. $\{a+b\sqrt{2} + ci+di\sqrt{2} : a,b,c,d\in\mathbb{Q}\}$?
2
votes
1answer
18 views

Potential Frobenius automorphism question

Let $F$ be a finite field of characteristic $p$ of size $p^n$ for $n \ge 1$ with the base field $K \cong Z_p$. I'm attempting to prove that the map $\phi: F → F$ sending $u$ to $u^p$ for each $u \in ...
1
vote
1answer
43 views

homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}$ correspondence theorem question

I am looking at the homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}$ that sends $x$ to $1$. I need to explain what the Correspondence Theorem when applied to this map says about the ideals of ...
0
votes
0answers
20 views

[KL:L]<[K:K inter L] [duplicate]

I'm asked to find two extension fields of a field $F$, such that $K/F$ is normal, $L/F$ is algebraic and $[KL:L] < [K:K \cap L]$. The first part of the exercise says that if either $K$ or $L$ is ...
0
votes
1answer
29 views

Field of fractions of integral extension is an algebraic extension [duplicate]

Let $A\subset B$ be an integral extension. If $F$ and $E$ are the fields of fractions of $A$ and $B$, respectively, I want to show that $E$ is an algebraic extension of $F$. I know that since $A ...
0
votes
2answers
42 views

The splitting field of $x^{3}-2$ over $\mathbb{Q}$ and its degree.

The roots of $f = x^3 -2$ are $\{2^{1/3}, a, a^2\}$, where $a = \frac{-1+\sqrt{3}i}{2}$. So let $E$ be the splitting field of $f$ over $\mathbb{Q}$, then $E = \mathbb{Q}(2^{1/3}, a)$. Now I attempt ...
2
votes
1answer
50 views

$\mathbb{Z}_p$ necessarily realised as galois group of characteristic $p$ field?

Question I want to ask is practically precisely what's in the question but I will restate to make it clearer. Suppose $k$ is a field of characteristic $p$ which is not algebraically closed. Then we ...
2
votes
0answers
19 views

Automorphisms of field generated by two coprime elements

I would like to know if the follwing statement is true: Let $F$ be a field and let $a,b$ be algebraic over $F$ with coprime degrees $m$ and $n$, respectively. Suppose $F(b)/F$ is normal. Letting $K = ...
1
vote
1answer
29 views

Field automorphisms of extension generated by two coprime algebraic elements.

Let $F$ be a field and let $a,b$ be algebraic over $F$ with $[F(a) : F] = n$ and $[F(b) : F] = m$ coprime. Let $\sigma \in \textrm{Aut}(F(a,b)/F)$. Is it true that $\sigma(F(a)) = F(a)$ and ...
5
votes
1answer
50 views

I know Galois theory is used to study fields using properties of groups. Is it ever used to study groups using properties of fields?

More specifically, are there any results in pure, abstract group theory that are most easily proved using Galois theory?
0
votes
0answers
35 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
0
votes
3answers
42 views

How can it be shown that for some prime p, $\mathbb{Q}$[$\sqrt{p}$, $\sqrt[3]{p}$] = $\mathbb{Q}$[$\sqrt[6]{p}$]?

I was told to consider the degrees but I'm not sure how the degrees of the polynomial so can help me here.
2
votes
0answers
21 views

On the restriction of field homomorphisms to subfields

Let $F/L/K$ be field extensions with $L/K$ finite. Let $H=\text{Hom}_K(L,F)$ be the set of field homomorphisms $L\rightarrow F$ that fix $K$. Take $\alpha\in L$, and let $\lbrace ...
1
vote
3answers
36 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
0
votes
2answers
43 views

how can one show that $\mathbb{Q}$($\sqrt{3}$, $\sqrt[3]{3}$, $\sqrt[4]{3}$, …) is algebraic but not finite dimensional?

The fact that this extension is infinite seems almost obvious and this is what makes it difficult to prove that the extension is algebraic. I would be able to do it for a finite case by identifying ...
0
votes
0answers
21 views

How to calculate the discriminant of a cubic equation easily

I'm trying to show the degree of the splitting field of a cubic polynomial with a zero quadratic term is related to the discriminant of the polynomial. On this process, i am trying to find the product ...
-1
votes
1answer
31 views

simple algebraic extensions with the same minimal polynomial [closed]

I can't see why $ji^{-1}$ is the identity on $K$, could someone explain please