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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2
votes
2answers
74 views

The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
0
votes
3answers
25 views

Ring Extension: Mapping: $ \mathbb Q[\sqrt d] \rightarrow \mathbb Q$

Show that the Norm: $\mathbb Q[\sqrt d] \rightarrow \mathbb Q, (r+s\sqrt d) (r-s\sqrt d) = r^2 - ds^2$ is multiplicative, i.d. $N(xy) = N(x)N(y)$ How to show it without computing? (I tried to do it ...
0
votes
1answer
31 views

A problem about degrees of minimal polynomials for two arbitrary elements in an extension field

I'm struggling to come up with a reasonable proof for the following problem: Suppose $E$ is an extension field of a field $K$ and that $a$ and $b$ are algebraic elements in $E$. Show that the ...
-2
votes
2answers
39 views

Extension fields, and their cardinality and roots

I have no idea how to begin answering this question. My notes do not help. Let $f(X) = X^3$ + $X + 1 ∈ \mathbb Z_5[X]$ ($\mathbb Z_5$ denotes the integers mod 5). Let $E=\mathbb Z_5[X]/(f(X))$. ...
1
vote
1answer
31 views

Show that $f$ is the minimal polynomial of $u$

Let $u$ be a root of $f=x^3-x^2+x+2\in \mathbb{Q}[x]$ and $K=\mathbb{Q}(u)$. Prove that $f=m_\mathbb{Q}(u)$. I have no idea how to approach this problem. Should I prove that $f$ is irreducible ...
0
votes
2answers
38 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated ...
0
votes
1answer
16 views

Unclear explanation of solution again;field extension

The solution sheet assumes additional knowledge than what is provided, which annoys me; I don't understand this. Here's the problem $L:K$ is a field extension. If $\alpha,\beta \in L$ is ...
1
vote
1answer
20 views

why aren't finite fields of prime characteristic algebraically closed?

How can this be proven? I know that if a field has a prime characteristic, any element of the field, say $a$. will satisfy the following equation: $ap = 0$, where p is the prime characteristic of ...
1
vote
0answers
33 views

Since field extensions are vector spaces over ground fields, how is linear algebra used to study them?

I've already taken a semester of Galois theory, so I'm not asking about anything that would be covered there, like the tower theorem. Specifically, multiplication by an element of a field extension is ...
3
votes
2answers
49 views

Galois group of splitting field over $\mathbb{Q}$

Describe the structure of the Galois group of the splitting field $L$ of the polynomial $X^4-7$ over $\mathbb{Q}$ I believe that $L=\mathbb{Q}(\sqrt[4]{7}, i)$ is the splitting field of the ...
2
votes
1answer
26 views

Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
2
votes
0answers
44 views

Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
1
vote
0answers
31 views

The tower of ramification indices

Let $K\subset L\subset M$ be an extension of number fields. Let $R\subset S\subset T$, be their algebraic integers rings, respectively. Suppose that $P\subset R$, $Q\subset S$, $U\subset T$ be a prime ...
0
votes
1answer
30 views

Prove a field - trouble with defining basic operations.

I'm certain this is a fairly easy question, but my algebra is rusty and I'm doing this as a part of a bigger proof. I'm stating that, if $\Bbb K$ is a field and $\Bbb K'$ its prime subfield, then 1) ...
0
votes
0answers
19 views

When are composite extensions isomorphic?

Let $E$ and $F$ be two totally complex finite extensions of $\mathbb{Q}$, let $\sigma_i \, :\,E \rightarrow \mathbb{C}, i\in I$ and $\tau_j \,: \,F \rightarrow \mathbb{C}, j\in J$ denote all their ...
1
vote
1answer
120 views

Splitting field of an irreducible polynomial of degree four [closed]

Suppose $f$ is the irreducible polynomial $x^4+x+1$ in $\mathbb{Q}[X]$ and we will call $E_f$ the splitting field of $f$ (which is a subfield of $\mathbb{C}$). Suppose $\alpha$ is a complex root of ...
0
votes
2answers
21 views

Abstractly constructing splitting fields

I have a series of exercises where I have to determine the degree of various splitting fields. I am freely using the following observation, which I feel is intuitively true, but I am asking here to ...
1
vote
1answer
28 views

An algebraic element $a$ in a field extension $K/F$ satisfies $a^{q^m}=a$

Let $F$ be a field with order $q$ and characteristic $p$. Show that if $a$ is an algebraic element over $F$ in the extension $K$, then $a^{q^m}=a$ for some $m$. I have shown that the order of the ...
0
votes
0answers
25 views

How can one show algebraically that an angle is constructible?

For example an angle of 30 degrees. I know that geometrically I can obtain the entire 30-60-90 triangle using the standard tools (compass, straightedge and unit length) and by performing iterations. ...
1
vote
0answers
23 views

$f(x) = x^2 + bx + a$ irreducible over $\Bbb F_p$ (finite field of $p$ prime elements) iff $(b^2 - 4a)^{\frac{p-1}{2}} = -1$ in $\Bbb F_p$

My attempt started as follows. I know that for $f$ to be irreducible, $D = b^2 - 4a$ is not a square in $\Bbb F_p$ (ie $(\frac{D}{p}) = -1$). I also know that $D^{p-1} = 1$, so I see $\sqrt{(D^{p-1})} ...
1
vote
0answers
34 views

Structure of Galois group

Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$ and $K|_F$ be the splitting field of $f(x)$. I want to prove the statement that if $G = \text{Gal}(K|_F)$ is cyclic then $[K:F] ...
1
vote
1answer
34 views

$\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of degree $a$

Let $r\geq1$ and $p\not\mid r $ prime, where $p\in(\mathbb{Z}/r\mathbb{Z})^*$ has order $a$. How do I prove that $\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of ...
0
votes
1answer
25 views

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$?

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? I was thinking about polynomial space and complexe space ...
2
votes
2answers
47 views

Proving $f(x)$ is not a square in $k[x]$

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
1
vote
1answer
45 views

Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
5
votes
5answers
105 views

Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$

Show that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ and find all $w\in \mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ such that $\mathbb{Q}(w)=\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$. It ...
0
votes
0answers
25 views

If $L=K(a_1,…,a_n)$ can we find an irreducible polynomial in $K[x]$ s.t. $p(a_i)=0$ for all $i$?

I have the following question that I can't prove or find a counterexample for. Let $K$ be a field and $L$ a finite field extension of $K$ so that we can write $L=K(a_1,..,a_n)$ where all $a_i\in ...
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
39 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
168 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
17 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
40 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= ...