5
votes
1answer
51 views

$F/K$ algebraic and every nonconstant polynomial in $K[X]$ has a root in $F$ implies $F$ is algebraically closed.

Let $F/K$ be an algebraic extension of fields in characteristic zero. If $F/K$ is normal, and every nonconstant polynomial $f \in K[X]$ has a root in $F$, then $F$ is algebraically closed. This is ...
1
vote
1answer
20 views

Is a finitely generated extension of a real closed field also real closed?

Let $\widetilde{\mathbb{Q}}$ be the field of real algebraic numbers, and consider $\widetilde{\mathbb{Q}}(\pi)$. My question: is $\widetilde{\mathbb{Q}}(\pi)$ a real closed field? Bonus karma points ...
1
vote
1answer
64 views

Characterize elements in $\mathbb Q(\sqrt2,\sqrt[3]5)$

I'm studying algebra by Herstein's Topics in Algebra, 2ed, and stuck at Ex.5.1.6(b): In $\mathbb Q(\sqrt2,\sqrt[3]5)$ characterize all the elements $w$ such that $\mathbb Q(w)\ne \mathbb ...
0
votes
1answer
35 views

A question about separable extension

We say $K$ is separably generated over $k$ if there exists a transcendence basis $\{x_i,i\in I\}$ of $K/k$ such that $K/k(x_i,i\in I)$ is a separable algebraic extension. We say $K$ is separable over ...
6
votes
2answers
88 views

Euclid and finite fields

In 300 BC or so Euclid pointed out that if $S$ is any finite set of prime numbers then the prime factors of $1+\prod S$ are not in $S$, so that $S$ can always be extended to a larger finite set. Much ...
1
vote
1answer
59 views

If $X^{p^d}\equiv X\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$

Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
0
votes
1answer
15 views

Splitting Field of Cubic Polynomial Over the Rationals

I'm having a hard time wrapping my head around some of concepts Pinter's Abstract Algebra introduces about splitting fields (or root fields, as it calls them). Hopefully if I can be pointed in the ...
3
votes
1answer
35 views

Embedding Fields in Matrix Rings

Is well known that the field $\mathbb C$ of complex numbers can be embedded in the ring $M_2(\mathbb R)$ of matrices of order two over de reals. In fact, $\varphi :\mathbb C\longrightarrow M_2(\mathbb ...
0
votes
2answers
19 views

A question about the minimal polynomials of linear transformations in fields.

Let $K$ be a finite field extension of $F$. Let $\alpha\in K\setminus F$. Then multiplication by $\alpha$ is an F-linear transformation form $K\to K$. Let linear transformation be called ...
3
votes
1answer
50 views

Is $\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$ a simple extension?

Is the extension $$\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$$ simple? If so find the minimal polynomial and the basis for the extension.
0
votes
2answers
30 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
0
votes
1answer
29 views

$[K:F_1]=[K:F_2]$, are $F_1, F_2$ isomorphic?

$K$ is a field extension of field $F_1$, $F_2$, all of them are finite fields. There is no other conditions. I want to know is the assertion possible? Could you please give a brief explanation?
1
vote
0answers
27 views

Splitting field and intermediate fields

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find $(K: \mathbb{Q} )$ $G(K/ \mathbb{Q} )$ Find intermediate fields between $\mathbb{Q}$ and $K$. So the ...
0
votes
1answer
24 views

Completion of a metric space vs. field extension

Given a field $F$ and a metric $\mu$, is the completion of $F$ with respect to $\mu$ always a field? Additionally, is there an algebraic field extension of $F$ that is isomorphic to the completion of ...
1
vote
1answer
49 views

Show that a sequence of fields exists

I do not have a clue how to solve the following problem: Let $K\subseteq L$ be Galois extension of degree $p^n$, where $p$ is prime and $n$ is natural. Show that there exists a sequence of subfields ...
1
vote
0answers
32 views

Galois extension of two fields

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it? For fields $K\subseteq L,M\subseteq \bar ...
5
votes
1answer
35 views

Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
1
vote
0answers
20 views

Field extension of fraction field of polynomial ring modulo an ideal.

My apologies for the relatively long question, but I am trying to understand a step in a proof, which needs some preliminary explanation. Let $K$ be a field and $I$ a prime ideal of ...
0
votes
0answers
22 views

Show that $K_1K_2=K_1(K_2)$ is abelian. [duplicate]

Let $L/F$ be a field extension and $L/K_i/F$ with $K_i/F$ abelian. Show that $K_1K_2=K_1(K_2)$ is abelian.
1
vote
3answers
128 views

Questions about $\Bbb Q[\sqrt{p}]$ and $\Bbb Q(\sqrt{p})$

I studied this part where they talk about $\Bbb{Q}(\sqrt{2})$ and $\Bbb{Q}[\sqrt{2}]$ and I really start to get confused. Definitions: $$ \Bbb{Q}[\sqrt{2}] = \left\{ a + b \sqrt{2} \mid a,b \in ...
1
vote
4answers
50 views

Proving $\mathbb{Q}(\eta,i)=\mathbb{Q}(\xi)$ where $\xi=e^{\pi i/10}$ and $\eta=e^{2\pi i/5}$

Proving $\mathbb{Q}(\eta,i)=\mathbb{Q}(\xi)$ where $\xi=e^{\pi i/10}$ and $\eta=e^{2\pi i/5}$. All I have got to prove is $[\mathbb{Q}(\eta):\mathbb{Q}]=4$, $[\mathbb{Q}(\xi):\mathbb{Q}]=8$ and ...
0
votes
1answer
26 views

Help Understanding Field Extension/Linear Algebra Problem

Let $E = F(\alpha)$ be a simple field extension of a finite field F by an algebraic element $\alpha$. Thinking of $E$ as an $F$-vector space, define a linear transformation $$T:E\rightarrow ...
3
votes
1answer
97 views

Question about the number of roots of irreducible polinomials.

Suppose $F|K$ is a field extension and let $\alpha \in F$ be an algebraic element over $K$ and let $\beta \in K(\alpha)$. Let $f=Irr(\alpha, K)$, $g=Irr(\beta, K)$, $h=Irr(\alpha, K(\beta))$. Let $L$ ...
1
vote
2answers
107 views

Field homomorphism into itself

Suppose $F$ is algebraic over $\mathbb{Q}$ and $\varphi \colon F \to F$ is a homomorphism. Prove that $\varphi$ is an isomorphism. I came across this question here but I don't quite get why $F$ ...
3
votes
1answer
61 views

Minimum polynomial of $\sqrt{2} + \sqrt[3]{5}$ above $\mathbb{Q}$ (and a generalization)

I have found that $\alpha = \sqrt{2} + \sqrt[3]{5}$ is a root of $f(x) = x^6-6x^4-10x^3+12x^2-60x+17$. I don't know if this is the minimum polynomial of $\sqrt{2} + \sqrt[3]{5}$ above $\mathbb{Q}$. I ...
3
votes
0answers
148 views

About $Z_{p}[\sqrt{k}]$, when is it a field?

I give up. I'm new in the fields world, and I'm trying to give a sufficient and necessary condition for $\mathbb{Z}_{p}[\sqrt{k}]=\{a+b\sqrt{k}:a,b\in \mathbb{Z}_{p}\}$ to be a field ($p$ is a prime ...
1
vote
2answers
50 views

Field extensions and gcd

Let $L|K$ be a field extension and let $u, v \in L$ be algebraic elements over $K$ such that $[K(u):K]=n$ and $[K(v):K]=m$. Show that if $\gcd(m, n)=1$ then $Irr(v, k)$ is irreducible on $K(u)$. ...
1
vote
0answers
43 views

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \ldots$

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime ...
3
votes
4answers
74 views

$\mathbb{Q}(\zeta_7)$ subextension of degree $3$

Let $\zeta_7$ be a $7$-th primitive root of unity. Is there a way to determine a subextension of $\mathbb{Q}(\zeta_7)$ that has degree $3$, without making use of Galois theory stuff?
4
votes
1answer
51 views

Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial ...
1
vote
1answer
43 views

Multiplicative Property of the degree of field extension

According to Artin's Algebra, chapter 15, section 3, the mapping property of the degree of field extension is as follows: Let $F\subset K\subset L$ be fields. Then $[L:F]=[L:K][K:F]$, where $[K:F]$ ...
4
votes
1answer
47 views

A Galois Extension over a field of characteristic p

Hi! I've been having a bit of trouble with this question, namely the very last part. It's from a past paper for a Galois Theory course, in which I have an exam on Monday. Basically, I can show every ...
1
vote
3answers
41 views

Basis of field - polynomial

Going through some old exams in my abstract algebra course, and was a bit curious to how I should neatly approach this problem. Let $F=\mathbb{Z_5}[x]/\langle x^3+x^2+1\rangle$ a) Give a basis of F ...
0
votes
3answers
47 views

Prove that $\alpha_{1} ^k+ \alpha_{2} ^k +…+ \alpha_{n} ^k = n$ for $k=0$ and $0$ for $k = 1,2,…,n-1$?

For $n\geq 2$ let $\alpha_{1} + \alpha_{2} +.....+ \alpha_{n} $ be all the nth roots of unity over a field and the roots are not necessarily to be distinct. So we have to prove that $\alpha_{1} ^k+ ...
2
votes
2answers
31 views

A result on extension fields in linear algebra.

Let $F$ be a subfield of $E$, $A$ an element of $\mathcal{M}_F(m,n)$ and $b$ a vector in $F^m \subset E^m$. What is the easiest way to prove the following statement: if $Ax = b$ has a solution in ...
4
votes
2answers
96 views

Galois group - extend homomorphism to automorphism

Let $K \subset L$ be a finite Galois extension, $M$ a field with $K \subset M \subset L$ and $G := \text{Aut}(L/K)$. I want to show that if $\sigma \, \colon M \longrightarrow L$ is a ...
0
votes
1answer
14 views

There are finite distinct restrictions to a subfield

Consider the field extension $L\subseteq K\subseteq \mathbb C$ where $K/L$ is finite. I must show that the set $\{\sigma_{|K}\,:\,\sigma\in\operatorname{Gal}( \mathbb C/L)\}$ is finite, but I have ...
1
vote
0answers
33 views

simple extension with algebraic over the field

Assume that $F$ is infinite, that $v,w \in K$ are algebraic over $F$, and that $w$ is a root of a separable polynomial in $F[x]$. How will I be able to prove that $F(v,w)$ is a simple extension of ...
0
votes
0answers
71 views

Prove that if $a^2 + ab + b^2 = 0$ then $a = b = 0$?

We are given that $a, b \in F_{2^n}$ where $n$ is an odd +ve integer. Suppose $a^2 + ab + b^2 = 0$ then we have either $a = 2^n-b^2$ or $a+b = 2^n - b^2$. Which implies that $\sqrt{2^n -a} = +-b $ or ...
1
vote
3answers
68 views

Prove that every extension of a finite field is normal

In book by Roman 'Field Theory' it is written that it is straightforward that every extension of a finite field is normal. However I just cannot see it. Can you help me with this problem? Thank
0
votes
1answer
52 views

finite field extension and normality

If $K$ is an extension field of $F$ such that $[K:F]=2$. Then $K$ is normal? I know that if $[K:F]=2$ then $K=F(u)$ where $u$ is the root of $f \in F[x]$. But how do you prove that dimension $2$ ...
1
vote
1answer
34 views

Rupture field of $X^p+T$ equals its splitting field [closed]

Let $K$ be a field of prime characteristic $p$. Let $P(X)=X^p+T$ be a polynomial from $K(T)[X]$. $P$ is irreducible over $K(T)$ by Eisenstein criterion. Show that a rupture field of $P$ is also a ...
2
votes
1answer
43 views

Embedding a finite extension of $F(X)$ into a pure transcendental extension

If $F$ is any algebraically closed field, and $L \supset F(X)$ is a finite extension of the purely transcendental extension of $F$ of transcendence degree $1$, then can $L$ necessarily be embedded ...
0
votes
1answer
21 views

Basis for field extension by an algebraic element

Is was wondering if, given a field F with a known basis and an element b which is algebraic over that field, it is possible to construct explicitly a basis for F[b], the extension of F by b. Suppose, ...
2
votes
1answer
79 views

Why isn't $e^{\frac{2\pi i}{9}}$ an element of $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})$?

The question is in the headline above. I need to know this, because I want to show that $\mathbb{Q}(\sqrt[9]{2}, e^{\frac{2\pi i}{3}})/\mathbb{Q}$ is not a normal extension and it's quite obvious I ...
0
votes
2answers
70 views

Calculate the degree of a field extension

Find degree $[\mathbb{Q}(i,\sqrt{2}) : \mathbb{Q}]$ let $a = \sqrt{2}$ $a^2 = 2$ $\therefore a$ is a root of $q(x) = x^2 -2$, where $q(x)\in\mathbb{Q}(i\sqrt{2})[x]$ means degree of $a$ over ...
0
votes
1answer
37 views

Show that two field extensions are the same

Can you help me with showing that these two field extensions are the same: $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$ $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number. Thanks ...
1
vote
1answer
49 views

Find a subfield of $\mathbb{C}$ isomorphic to other field

Do you know, how I can find a subfield of $\mathbb{C}$ isomorphic to $F = \mathbb{Q}(\sqrt[3]{7})$ such that $F \nsubseteq \mathbb{R}$? I don't even have clue, how I should start. Thanks
0
votes
1answer
60 views

The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
0
votes
1answer
30 views

Convert from a field extension to an elementary field extension

I have a basic question about algebraic field extensions: How can I convert a multiple extension like $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to a single (elementary) field extension (like ...