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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$\mathbb{Q}(\sqrt2) $ is isomorphic to $\mathbb{Q}(\sqrt2 +2 )$

Show that $\mathbb{Q}(\sqrt2) $ is isomorphic to $\mathbb{Q}(\sqrt2 +2 )$, but is it more? Are these fields equal? $\mathbb{Q}(\sqrt2)=\{a+b\sqrt2 |a,b \in \mathbb{Q}\}$ ...
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28 views

Exercise about cyclic field extension

I am having hard time to solve following exercise. Let $\Omega$ be the algebraic closure of a field $k$. a) Suppose that every finite extension of $k$ is cyclic. Prove that it exists $\sigma \in ...
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1answer
36 views

Number of Galois conjugates

Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension. My question: Is the ...
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1answer
83 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
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1answer
21 views

L|K field extension and $deg(f)\nmid[L:K]$

Let L|K be finite field extension and $f \in K[X]$ is irreducible with $deg(f) > 1$. Show that, if $f\nmid[L:K]$ then f has no zeros in L. Is it true? For ex. $f=x^3+x$ and $[Q(\sqrt 5,i):Q]$. f ...
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2answers
90 views

Polynomials over a finite field

Let $\mathbb{F}_p$ be a finite field where $p$ is a prime. Consider the following set of polynomials over $\mathbb{F}_p$: $$G_n(p)=\{{x+a_2x^2+\cdots+a_nx^n\mid a_i\in \mathbb{F}_p}\}.$$ Is ...
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1answer
73 views

Galois Group of $x^4 - x^2 - 3$

Find the Galois Group of $x^4 - x^2 - 3$ This is a qual question. I don't know how to find the splitting field of this polynomial.
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2answers
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Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
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1answer
48 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
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1answer
25 views

Finding the subfields of the cyclotomic field of order $5$

This is part of an exercise from Hungerford's Algebra: Find all intermediate fields in the field extension $F_5/\mathbb{Q}$, where $F_5$ is the cyclotomic extension of $\mathbb{Q}$ of order $5$. ...
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1answer
19 views

Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff ...
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0answers
30 views

Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes of ...
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1answer
91 views

Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
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0answers
58 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
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2answers
42 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
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2answers
54 views

Naïve groups, fields and ideals

Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right ...
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2answers
63 views

Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote ...
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1answer
38 views

Reducing splitting field

If we have splitting field: $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{2-i\sqrt{6}}) $$ we can multiply these two zeroes and get $\sqrt{10}$ so we have $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
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2answers
48 views

Are the fields $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ and $\mathbb{Q}(\sqrt[7]{16})$ equal?

I have trouble with these field extensions. Is field $\mathbb{Q}(\sqrt[7]{16}+3 \sqrt[7]{8})$ equal to field $\mathbb{Q}(\sqrt[7]{16})$? We can $\sqrt[7]{16}+3 \sqrt[7]{8}$ express as ...
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1answer
35 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
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3answers
45 views

List the elements of the field $K = \mathbb{Z}_2[x]/f(x)$ where $f(x)=x^5+x^4+1$ and is irreducible

Since $\dim_{\mathbb{Z}_2} K = \deg f(x)=5$, $K$ has $2^5=32$ elements. So constructing the field $K$, I get: \begin{array}{|c|c|c|} \hline \text{polynomial} & \text{power of $x$} & ...
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1answer
43 views

Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
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2answers
63 views

Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
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1answer
66 views

What are the groups $\text{Hom}(F^\times\!, F^+\!)$ and $\text{Hom}(F^+\!, F^\times\!)$?

Background. Exercise 36 in Rose's A Course On Group Theory reads Prove that there is no field $F$ with $F^\times \cong F^+$. The problem is solved in characteristic $\ne 2$ by considering $-1$ ...
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1answer
63 views

Proof that every field $F$ has an algebraic closure $\bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $\bar F$. I ...
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0answers
38 views

How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any ...
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1answer
45 views

Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of ...
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2answers
57 views

How can we work with field extensions when our base fields aren't actually subfields?

I've been wondering this for a little while. Say we are working with the rational numbers $\mathbb{Q}$, and then we wish to talk about the extension fields $E$ of $\mathbb{Q}$, by which we mean the ...
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0answers
32 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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1answer
12 views

Splitting of primes terminology doubt

What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...
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2answers
37 views

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$.

How should I find Splitting Field of $x^3-2$ over $\mathbb Q$. **My try **: $x^3-2=(x-2^\frac{1}{3})(x^2+2^\frac{1}{3}x+2^\frac{2}{3})$ On solving I am getting the roots as ...
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1answer
39 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
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0answers
21 views

Galois closure uniqueness confusion

I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote) Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is ...
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1answer
52 views

From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
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1answer
29 views

Field extension of prime degree

Question: Let $L$ be the extension of the field $K$ such that $[L:K]=p$, where $p$ is a prime number, and $\alpha \in L$. Prove that $K(\alpha)=K$ or $K(\alpha)=L.$ Proof: From $$ \alpha \in L ...
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0answers
32 views

Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
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0answers
38 views

What does this theorem mean, exactly?

The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $F_p[X]$ of degree $d$ where $d$ runs through all the divisors of $n$. I don't even get the ...
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8 views

Question on separable degree

I asked a similar question before but I didn't get a satifying answer, so I'm posting it again. Let me first define terms: Def1 Let $E/F$ be an algebraic field extension and $\bar F$ be an ...
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1answer
78 views

If there are $k_1, k_2 \in K$ such that $K(\alpha + k_1\beta)=K(\alpha + k_2\beta)$ then $K(\alpha,\beta) = K(\alpha + c\beta)$ for some $c \in K$.

Here is the problem: "Let $K \subset M$ be a finite field extension, and $\alpha, \beta \in M$. Suppose there are $k_1, k_2 \in K$ are distinct and such that $K(\alpha + k_1\beta)=K(\alpha + ...
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34 views

What is the possible number (supremum) of subfields of $\mathbb{F}$?

Let $\mathbb{F}$ be field. it is a finite dimensional extension over $\mathbb{Q}$. So let $B=\{v_1, v_2, v_3, ... , v_n\}$ be a basis for $\mathbb{F}$ over $\mathbb{Q}$. From the finite dimension ...
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1answer
46 views

Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
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3answers
100 views

$\alpha \in \overline{\mathbb{F}}_q$ satisfying $\alpha^{q+1}+\alpha=-1$

Let $\overline{\mathbb{F}}_q$ be the algebraic closure of $\mathbb{F}_q$. Assume that $\alpha \in \overline{\mathbb{F}}_q$ satisfies at $$\alpha^{q+1}+\alpha=-1$$ Show that $\alpha \in ...
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1answer
26 views

Finding proper subfields

Let $\omega$ denote the cube root of unity such that $\omega\neq 1$. I want to find the subfields properly contained in $\mathbb Q(\sqrt[3]{2},\omega)$ and containing $\mathbb Q$ properly. Two of ...
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1answer
33 views

Relation between $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$

I have the following exercise in my homework: Are $\mathbb{Q}(2^{1/3})$ and $\mathbb{Q}(w2^{1/3})$ isomorphic, where $w = \textrm{cis}((2\pi)/3)$? Prove your answer. I think they are, but I'm ...
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5answers
70 views

looking for the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$ with $\alpha=\sqrt[3]{2}$

I have $\alpha=\sqrt[3]{2}$ and want to calculate the inverse of $3+2\alpha^2$ over $\Bbb Q(\alpha)$. There's a hint which tells me to look at the minimal polynomial $m_\alpha$ of $\alpha$ over $\Bbb ...
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1answer
32 views

Is the statement of the theorem correct?

I have been asked to prove this:: $f,g$ are polynomials over a field $F$ .Prove that if $f,g$ are relatively prime then $f,g$ have no common roots in any extension of $F$. But I wonder why is ...
4
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2answers
22 views

Non-algebraic subfield intersection

Construct an example in which a field $F$ is of degree 2 over two distinct subfields $A$ and $B$, but so that $F$ is not algebraic over $A\cap B$. I'm having trouble thinking of an explicit example ...
4
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1answer
50 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
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1answer
45 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
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1answer
45 views

How do I find the quotient field of $\mathbb{Z}[\sqrt{d}]$?

Our teacher said sometimes the quotient field is $\mathbb{Q}[\sqrt{d}]$ and sometimes it's $\mathbb{Q}[\frac{1+\sqrt{d}}{2}]$. How do we decide, or what are the conditions on $d$ which helps us to ...