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
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1answer
35 views

Is the degree of an infinite algebraic extensions always countable?

I guess this is right and try to prove it by using the fact that the polynomial ring $K[t]$ has a countable basis $1,x,x^2,\cdots$. But How to use this fact? Aside, if this statement is true. Is the ...
1
vote
1answer
24 views

Field Theory Problem in Beachy's Abstract Algebra involving field extensions and transcendental elements.

Let $\mathbb{F}=\mathbb{K}[u]$ where u is transcendental over $\mathbb{K}$. Show that if $\mathbb{K} \subsetneq \mathbb{E} \subseteq \mathbb{F}$ then u is algebraic over $E$. I'm guessing that I need ...
0
votes
1answer
22 views

Group under addition structure of a finite field of order 9 [on hold]

Suppose $(F,+,\cdot)$ is the finite field with 9 elements.Let $G=(F,+)$ and $H=(F\setminus\{0\},\cdot)$ denote the underlying additive and multiplicative groups, respectively. Then, (a) $G=(Z/3Z) ...
3
votes
3answers
38 views

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$

for $n > 1$ : odd , prove that $\Phi_{2n}(x) = \Phi_{n}(-x)$ Note: $\Phi_n(x)$ is the $n$th cyclotomic polynomial whose roots are the primitive $n$th roots of unity if n is odd then $-1$ cannot ...
3
votes
1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
1
vote
1answer
22 views

Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$? As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...
0
votes
1answer
13 views

Showing polynomial is irreducible over field containing roots of unity.

Given a field $F$ containing all the roots of unity I'm trying to show that $f(x) = x^p - \alpha^p$ is irreducible over $F$ (where $\alpha$ is not in $F$). It's clear that $f$ splits in $F(\alpha)$ ...
1
vote
0answers
10 views

What is a separable closure? And why the standard definition of algebraic closure is defind as so?

Instead of defining it as a subfield of algebraic closure, what would be the natural way to define it? Here's what I figured out: Let $F$ be a field and $E$ be an extension field of $F$. ...
1
vote
1answer
24 views

Polynomial Rings

I know that an Euclidean ring is a principal ring and this last is a factorial ring. I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a ...
0
votes
2answers
38 views

Finite fields, characteristics and the Fundamental Homomorphism Theorem

I am trying to make sense of this proposition. I am fine with part (a), for part (b) however, can you explain what the computation proves? Can you not verify a homomorphism by checking the 3 ...
0
votes
0answers
24 views

Purely transcendental proper extension not algebraically closed? [on hold]

I'm having trouble proving this Dummit and Foote exercise: Prove that a purely transcendental proper extension of a field is never algebraically closed.
3
votes
1answer
17 views

Is normal extension algebraic?

Let $E/F$ be a field extension. Then $E/F$ is called normal iff there exists $\mathscr{A}\subset F[X]\setminus\{0\}$ such that $\forall f\in\mathscr{A}$, $f$ splits over $E$ and ...
4
votes
2answers
36 views

What will the underlying group of a field be isomorphic to?

Let $(F,+,.) $ be a finite field with 9 elements .,Let $G=(F,+)$ and $H=(F\setminus \{0\},.)$ denote the underlying additive and multiplicative groups .Thenwhat will $G$ and $H$ be isomorphic to:? WE ...
0
votes
2answers
29 views

How to show $x,y,z \in A$ - Functions, Combinatorics

If $A \subseteq \{1,2,3,4,5,6\}$, how to show that for every $A$ there are $x,y,z \in \{1,2,3,4,5,6\}$, where $x,y,z$ can also be the same or at least not different from each other, and the following ...
1
vote
2answers
45 views

Field extension over $\mathbb{Q}$

I am given a subfield $E$ of $\mathbb{C}$ and asked to show that $[E : \mathbb{Q}] \le 10$ when every element of $E$ is a root of a polynomial in $\mathbb{Q}[x]$ of degree $10$. But I don't think ...
0
votes
0answers
20 views

Relations between galois group of polynomial and its factors

$f = f_1\dots f_n$ where $f_i$ is irreducible and distinct. What can i say about $\operatorname{Gal}(f/\mathbb{Q})$ if i know $\operatorname{Gal}(f_i/\mathbb{Q})$ for any(some) $i$.
4
votes
0answers
59 views

Galois group of $x^6-5x^3+6$

Let $f = x^6-5x^3+6$. I want to determine $\operatorname{Gal}(f/\mathbb{Q})$ without some group theory tricks (like Sylow's theorems) and without reduction $\bmod p$. Let $L_f = ...
2
votes
2answers
30 views

Galois group of $x^6-9$

$f = x^6-9 = (x^3-3)(x^3+3)$ Let $L_f$ be splitting field therefore $L_f = \mathbb{Q}[\sqrt[3]{3},e^{\frac{2\pi i}{3}}]$, $[L_f:\mathbb{Q}] = 9$. Also $Gal\space x^3±3/\mathbb{Q} = S_3$ and $Gal\space ...
0
votes
0answers
17 views

Whether a given collection is a set or not [duplicate]

I originally knew that a set is a concept that has no definition. However, today, in abstract algebra class, the professor told us that the collection of all fields E such that E/F is an alegbraic ...
0
votes
1answer
19 views

Compositum of normal extensions is a normal extension

I'm trying to prove that if $ F \subset K, F \subset M $ are normal extensions, $ K,M \subset E $, then $ KM$ is also a normal extension of $ F $. I tried using the fact that $ F \subset KM $ is a ...
0
votes
3answers
42 views

Minimal polynomial of $\sqrt{2} + \sqrt{3}$ over $\Bbb{Q}(\sqrt{6})$

I have to find the minimal polynomial of $\alpha = \sqrt{2} + \sqrt{3}$ over $\Bbb{Q}(\sqrt6)$. $\alpha^{2} = 2 + 2\sqrt6 + 3$ so $f(X) = X^{2} - 5 - 2\sqrt6$ is a polynomial where $f(X) \in ...
0
votes
2answers
32 views

Degree of extension

How to find the degree of $\mathbb Q\left(\sqrt2+\sqrt[3]2\right) $ over $\mathbb Q\left(\sqrt2\right)$ ? I know how to find $\mathbb Q\left(\sqrt2\right)$ over $\mathbb Q$. But i am confused in ...
2
votes
1answer
22 views

Determine the Galois group of $ F(x^5) \subset F(x) $

I'm rather new to Galois theory and have been given this exercise: Suppose $ F $ is respectively equal to $ \mathbb{Q}, \mathbb{C}, \mathbb{F}_5 $ (the third one is just the 5-element field). My task ...
0
votes
0answers
18 views

Subfields of splitting field of $x^3+x+1$

$f = x^3+x+1, L_f$ - splitting field of $f$. Discriminant $D = D(f) = -31$ therefore $\deg L_f/\mathbb{Q} = 6$ and $Gal L_f/\mathbb{Q} = S_3 $. I want to find all subfields of $L_f$. $L_f = ...
1
vote
0answers
14 views

Obtaining formula for roots of cubic equalation

$f = t^3+pt+q \in \mathbb{C}(p,q)$ I want prove that splitting field of $f$ is $$\mathbb{C}(p,q)[D,x]$$ $\mu_x = t^3-a$(over $\mathbb{C}(p,q)[D]$),$\mu_D = t^2-b$(over $\mathbb{C}(p,q)$). I think that ...
3
votes
1answer
85 views

Show that $\mathbb{Q}(\sqrt p \mid p \text{ is prime})$ is an algebraic and infinite extension of $\mathbb{Q}$

Show that $\mathbb{Q}(\sqrt p \mid p \text{ is prime})$ is a algebraic and infinite extension on $\mathbb{Q}$. Well, if i consider for every $p$ prime, the polynomial $p(x)= x^2 - p$, then $p(x)$ is ...
1
vote
1answer
21 views

Question on separable field extenions

Hi I was given this question which I cannot express myself mathematically on so would indeed like the help and appreciate it I am given $ K/F $ is a finite field extension. I am required to show that ...
1
vote
1answer
16 views

Prove $\sigma_g(x) \in Aut(R(x)/R)$

Let $R$ be a field and let $R(x)$ be the field of rational functions in $x$ whose coefficients are in $R$. Let $g = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right) \in ...
3
votes
3answers
52 views

In a commutative ring with identity, if $p$ is irreducible, is ($p$) a maximal ideal?

In a Euclidean Domain, $D$, if we mod out by an irreducible, $p$, we get the field $D/(p)$. I can see that this follows since we are going to be able to write $1$ as a linear combination of $p$ and ...
3
votes
1answer
51 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...
0
votes
1answer
33 views

Let E be an extension of a field F of degree 2. Show there is an element with $\beta = x^2 - \alpha$

Let E be an extension of a field F of degree 2. Show there is an element with $\beta = x^2 - \alpha$. I wanna know if my approach is correct. If it's an extension of degree 2, then $F(\alpha)$ ...
1
vote
1answer
26 views

Exercise: splitting field, showing that it splits

I need help with this exercise: Let $\alpha$ be a zero of $x^3+x^2+1$ in $\mathbb{Z}_2$. Show that $x^3+x^2+1$ splits in $\mathbb{Z}_2(\alpha)$. [Hint: There are eight elements in ...
1
vote
1answer
26 views

a question about field theory and polynomials

Hello all I was given this question in my field theory class on which I would certainly appreciate the help: I am given a field F of characteristic p ($ ch(F) > 0 $) and this polynomial $ f(x) = ...
1
vote
1answer
20 views

finding matrix represention for linear transformation for field extension

need some clarification. given an extension field K over F with F-linear transformation, for $\alpha \in K$, $f_\alpha(k) = \alpha \cdot k$ i.e. multiplication on the left. I need to find the ...
0
votes
3answers
51 views

Question about field extentions?

if $\mathbb{Q}(\sqrt{3}) $ can be looked at as the field of rational numbers with $\sqrt{3}$ appended to it, and can be furthermore looked at like $\mathbb{Q}[x]/x^2 - 3$ what does a field extention ...
1
vote
2answers
23 views

Finding the conjugates, why can they argue this way?(exercise)

In one exercise I am supposed to find the conjugate of $\sqrt{2}+i$ over $\mathbb{Q}$. I found the answer by finding irr$( \sqrt{2}+i,\mathbb{Q})$, and then solving the polynomial finding all the ...
2
votes
2answers
55 views

Elements in a field

Prove that there exists a field with $16$ elements. I know that there is a theorem that states that a finite field can only have $$ p^k $$ Where $p$ is a prime and $k$ is any positive integer. But ...
-2
votes
3answers
58 views

Show that $\sqrt{5} \notin \mathbb{Q}(\sqrt{3})$

Show that $\sqrt{5} \notin \mathbb{Q}(\sqrt{3})$. I don't even know where to start. I can't find references to this in my textbook anywhere. I feel like the notation came out of nowhere.
1
vote
1answer
54 views

Why can't Eisenstein Criterion be used for certain polynomials (to show that it's irreducible over $\mathbb{Q}$)?

Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$? I mean even if we were to apply Eisenstein here, there doesn't exist a prime ...
-2
votes
3answers
49 views

Notation Question(Abstract Algebra)

what does $\mathbb{Q}(\sqrt{3})$ mean?
2
votes
1answer
28 views

What is the automorphism group of the field $\mathbb{Z} /p\mathbb{Z}(t)$?

Here, $t$ is transcendental over $\mathbb{Z} /p\mathbb{Z}$. How big is this group? What are its elements? Is for example the map $t \to -t$ an automorphism?
1
vote
1answer
42 views

Field extensions and algebraic elements

Can somebody explain why taking beta gives $K(\beta)$ as a subspace of $K(\alpha)$?
2
votes
2answers
43 views

Field that is a subfield of own of its subfields

Let $K$ and $L$ be fields. We have homomorphisms $f: K \to L$ and $g: L \to K$. Are $K$ and $L$ necessarily isomorphic?
0
votes
1answer
42 views

Automorphisms (in the context of Galois Theory)

Can someone give an explanation of what an automorphism is, in the context of Galois Theory? I keep thinking it is the set of maps which send roots of polynomials to their conjugates but I feel that ...
3
votes
1answer
55 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
2
votes
1answer
46 views

Does all splitting fields have characteristic 0?

Does all splitting field have characteristic 0? This may be a bad question, but I am wondering because the author of a book I am reading summarises a lot of properties when he is about to start with ...
4
votes
6answers
66 views

Prove this polynomial falls within $\mathbb R[x]$

[ The problem below is from Yao Musheng (姚慕生), Wu Quanshui (吴泉水), Advanced Algebra (高等代数学) Ed $2$, Fudan University Press, page $207$. ] Suppose $f(x)\in \mathbb C[x]$. If $\forall c\in \mathbb ...
0
votes
2answers
18 views

Surd-like trinomials form a field

This is a problem from Artin's book "Algebra". In the fifth miscellaneous problem of the chapter "Vector spaces", he has asked to prove that: If $\alpha$ is a cube root of $2$, then the real numbers ...
0
votes
0answers
34 views

Can it be proved that this extension is algebraic?

Assume that we have a field F, an extension field E of F, and both of them are contained in the algebraic clousure $\overline{F}$. Let E have the property that every automorphism of $\overline{F}$ ...
1
vote
1answer
20 views

Extension E/K such that E/F is a splitting field

The question asks us to prove that there is an extension $E/K$ such that $E/F$ is a splitting field of some polynomial $f(x) \in F[x]$ where $K/F$ is a finite extension. I'm not really sure how to ...