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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0
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1answer
20 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
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3answers
43 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
1answer
34 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 ...
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0answers
19 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
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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
94 views

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

Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is a algebraic and infinite extension on Q. Well, if i consider for every p prime, the polynomial p(x)=x^2−p, then p(x) is in Q(p√∣p is ...
1
vote
1answer
22 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
17 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
53 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
54 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
40 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
21 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
53 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
57 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
59 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 ...
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3answers
49 views

Notation Question(Abstract Algebra)

what does $\mathbb{Q}(\sqrt{3})$ mean?
2
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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
43 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
44 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
59 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
49 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
35 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
21 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 ...
1
vote
0answers
17 views

Dimension of compositum of two fields, one of them Galois.

Let $L, K, F = L \cap K$ be fields such that $[L:F] , [K:F] < \infty$. $LK$ is defined as the compositum of the two fields(shortest field containing $K$ and $L$ in some algebraic closure). Also, ...
1
vote
2answers
54 views

Question in Algebraic closed field

I'd like to know how to prove algebraic numbers form a field, i.e, if $a,b$ are 2 algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...
1
vote
1answer
28 views

Primitive element theorem, simple extension

Let $X$, $Y$ be indeterminates over $F_2$, the finite field with 2 elements. Let $L = F_2(X, Y )$ and $K = F_2(u, v)$, where $u = X + X^2$, $v = Y + Y^2$. Explain why $L$ is a simple ...
0
votes
1answer
22 views

Proof help: why is the constructed field a splitting field?

Here is my books definition of a splitting field: Note that it uses the word: smallest: In the last converse part of this theorem. I see that the field E created is a field that contains F(this is ...
7
votes
5answers
85 views

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ the same as $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I mean, $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ can be viewed as: $\mathbb{Q}[\sqrt{2}][\sqrt{3}]$, as polynomials of ...
0
votes
3answers
44 views

Showing that $\mathbb{Z}_N$ is a field if $N$ is prime

I know that $N$ being prime is a necessary and sufficient condition for $\mathbb{Z}_N$ to be a field. I know how to prove that it's necessary but I'm not sure how to prove that this is a sufficient ...
0
votes
0answers
46 views

Galois extension over power series fields

Let $K$ be a field, and $L$ be an algebraic extension of $K$. I think it is known that if $T$ is a finite extension of $K((X))$, then $T$ is complete with respect to the $X$-adic valuation, hence if ...
1
vote
1answer
26 views

Showing that $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ irreducible over $\mathbb{Q}(\sqrt[4]{5})$

I am unsure how to show that the polynomial $f(x) = \frac{1}{2} x^3 + 3x^2 -\frac{5}{3} x - 5$ is irreducible over $\mathbb{Q}(\sqrt[4]{5})$. If it were reducible, it would have a root in ...
3
votes
1answer
30 views

Algebraic or not algebraic extension?

Suppose $F^{\prime}/F$ is an algebraic extension of fields, and that $F^{\prime}$ is a finite field extension of $K^{\prime}(x^{\prime})$, where $x^{\prime}$ is transcendental over $K^{\prime}$, and ...
8
votes
0answers
67 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
5
votes
0answers
69 views

algebraically closed fields of characteristic 0 and $\mathbb{C}$

Let $k$ be an algebraically closed field of characteristic 0. Then I've heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding ...
0
votes
1answer
59 views

If $d \equiv 1 \pmod 4$, is $\mathbb Q[\sqrt d]$ the field of fractions of $\mathbb Z\left[\frac{1+\sqrt d}{2}\right]$?

If $d \equiv 1 \pmod 4$, is $\mathbb Q[\sqrt d]$ the field of fractions of $\mathbb Z\left[\frac{1+\sqrt d}{2}\right]$? Is $\mathbb Q\left[\frac{1+\sqrt d}{2}\right]$? I am confused about quadratic ...
0
votes
2answers
26 views

If $ f \in \mathbb{Q}[x] $ is irreducible and has a root in $ \mathbb{Q}(\sqrt{2}, i) $, then it splits

I'm trying to find a solution for the following problem: let $ f \in \mathbb{Q}[x] $ be irreducible. Suppose $ f $ has a root in $ \mathbb{Q}(\sqrt{2}, i) $. Prove that $ \deg f \in \{1,2,4\} $ and ...
5
votes
2answers
40 views

Degree of a splitting field over $ \mathbb{Q}$

I'm trying to solve the following problem: Let \begin{equation*} f(x) = x^4 - 2x^2 - 2 \in \mathbb{Q}[x] \end{equation*} and $ E $ be its splitting field. What is the degree $ [E: \mathbb{Q}] $? ...
1
vote
0answers
22 views

Action of the group ring $\mathbb{Z}[\text{Gal}(K/\mathbb{Q})]$ on the field $K$

Let $K$ be an algebraic number field, let $G$=Gal($K/\mathbb{Q}$). Let $\mathbb{Z}[G]$ be the group ring, or the set of formal sums $$\left\lbrace\sum a_i\sigma_i : a_i\in \mathbb{Z}, \sigma_i \in ...
1
vote
0answers
18 views

Notion of Separability

Notion of Separability An irreducible polynomial $f\in F[X]$ is separable, if $f$ has no repeated root in a splitting field, if $f$ is not necessarily irreducible, then we call $f$ separable, if ...
4
votes
2answers
121 views

Is the intersection of finite index subfields finite?

Suppose that $K$ and $L$ are two fields contained in some larger field, and let $KL$ denote the smallest subfield of the ambient field containing both of them. If $KL$ is a finite extension of both ...
0
votes
0answers
21 views

unramified extension of valued fields

I came across the following exercise: Let $M$ be a valued field with subfields $E$ and $L$, and suppose that $L$ is finite over some field $K\subseteq L\cap E$. Show that $EL/E$ is unramified if ...
1
vote
2answers
50 views

Why is $\mathbb{Q}$ left fixed?

In this example I get that 1 is left fixed, because every multiplicative element of an isomorphism is left fixed?, but why is $\mathbb{Q}$ left fixed as a consequence of this?