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 topic....

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normal extension of $\mathbb{Q}$

We define an algebraic extension $K/F$ to be normal if every irreducible $f \in F[x]$ with one root in $K$ splits in $K[x]$. Now, in my lectures it was stated that $\mathbb{Q}(i)$ is a normal ...
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2answers
58 views

Prove that $\mathbb{Q}[x] / \mathbb{Q}[x] (x^{3} - 2)$ is a field and find the inverse of of $[x - 3]$

I'm not sure how to prove the first part other than to go through all the axioms. As for the second, I know I need to find an element such that when it is multiplied to $[x - 3]$, it results in $[1]$...
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1answer
379 views

What are the subfields of $\mathbb{Q}(\sqrt{3},\sqrt[7]{5})$?

I'm trying to find the subfields of $\mathbb{Q}(\sqrt{3},\sqrt[7]{5})$. This is easily seen to be a degree $14$ extension of $\mathbb{Q}$. I found that there is a unique subfield of degree $2$ over $...
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1answer
65 views

Definition of a field homomorphism

Given a field $F$ of characteristic zero, say $F=\mathbb{R}$, what is the minimal requirement for a function $\mu:F\to F$ to be a field homomorphism? (Do we need to require two axioms, one for ...
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2answers
170 views

Basic Question on Field Theory - Hopefully to Illustrate Several Concepts

I am just beginning to study fields and for whatever reason am finding their presentation to be completely baffling - moreso than I think anything I have ever studied. I am reading out of chapter 21 ...
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76 views

Elementary Field Theory: Extension Field of Degree 2

I'm trying to do/understand the following exercise: "Let $E$ be a finite extension of a field $F$. If $[E:F] = 2$, show that $E$ is a splitting field of $F$."* Background: Just beginning ...
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1answer
70 views

Field Theory, Factor Ring, Polynomials

I have the following problems: (1) Let $g=X^2+\overline{4}$ and $h=X^2+\overline{2}$ be polynomials in $(\mathbb{Z}/\mathbb{Z}7)[X]$. $L$ and $K$ are the splitting fields of $g$ and $h$ over $\mathbb{...
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66 views

How does one characterise the reals without points?

The Real number line is defined upto unique isomorphism in the category $Fld$ as a Dedekind complete ordered field. One can view the Reals geometrically with its usual topology. In pointless ...
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2answers
566 views

Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
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1answer
39 views

Splitting Fields and Non-constant Polynomials.

I had a question I was stuck on: Let $p(x)$ be a non-constant polynomial of degree $n$ in $F[x]$. Prove that there exists a splitting field $E$ for $p(x)$ such that $[E : F] \leq n!$. My start: By ...
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2answers
74 views

Do conjugate fields have the same maximal order?

Suppose that $\alpha,\beta\in K$ are conjugates elements, i.e. zeros of an irreducible polynomial over $\mathbb{Q}$. Then we know that the fields $K_{1}=\mathbb{Q}(\alpha)$ and $K_{2}=\mathbb{Q}(\...
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1answer
258 views

Simple radical extension

Let $F/K$ be a Galois (finite) extension with solvable group. Must $F$ be a simple radical extension of $K$? or at least have an intermediate field which is a simple radical extension? If $F/K$ is ...
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0answers
53 views

Galois group of composite of Galois extensions

I'm reading through the proof in Dummit and Foote p. 593 that $$\operatorname{Gal}(K_1K_2/F) \cong H := \{(\sigma, \tau) \in \operatorname{Gal}(K_1/F) \times \operatorname{Gal}(K_2/F) \mid \sigma|_{...
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0answers
84 views

Characterizing $\operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q})$ for $p_i$ primes?

For what $n$ is $$ \operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q}) $$ known, where the $p_i$ are primes? By Kummer theory, I think that $$ \operatorname{Gal}(\mathbb{Q}(\sqrt{...
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2answers
545 views

Prove subset of Field is a subfield

Let $K$ be a field of characteristic different from 2, $F$ an algebraic extension of $K$ and $L$ a subset of $F$ with the following properties: $L$ contains $K$, $L$ is a K-vector space, $\forall v \in ...
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1answer
286 views

Characteristic of a field extension

Let $\mathbb{F}_p$ be a field. Is it correct that the field extensions are $\mathbb{F}_{p^n}$, with $n \in \mathbb{N}$? And does the characteristic of every $\mathbb{F}_{p^n}$ equal $p$?
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97 views

Field Extensions and Degree.

I had a question, I was just wondering about something. So there's a question in my textbook that asks for the degree of this field extension for:$$\mathbb{Q}(i, \sqrt2 + i, \sqrt3 + i)$$ I can ...
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1answer
193 views

Proof of theorem on irreducible polynomial

Quoting from Herstein's Abstract Algebra (3rd edition): Lemma 6.4.1. If $q(x)$ in $\mathbb{Z}_p[x]$ is irreducible of degree $n$, then $q(x) \mid (x^m- x)$, where $m = p^n$. Proof. By ...
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1answer
94 views

Given a finite field $\mathbb{F}$ of order $2^n$, how to construct a field of order $2^{2n}$?

Specifically, I would like to construct a field of order $2^{2n}$ with elements being $2\times2$ matrices whose entries are elements of $\mathbb{F}$. I know the complex numbers can be represented as $...
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1answer
85 views

$f(x) \mid g(x) \iff g(x) \in \langle f(x) \rangle$. Isn't this trivial?

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
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0answers
69 views

A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[t_1 \cdots t_n] \twoheadrightarrow F$

Does this make sense as an alternative definition for a finitely-generated field extension?: A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[...
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1answer
40 views

Unicity of order/positives in ordered field

I was remembering some stuff from my calculus course, and I got to the following question: By a field order, I mean a total order $\leq$ on a field $\mathbb{F}$ which is invariant by addition and by ...
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1answer
736 views

Why is the Galois closure of $K/F$ the composite of the Galois conjugates of $K$?

Suppose $K/F$ is a field extension with Galois closure $L$, and let $G=\operatorname{Gal}(L/F)$. Why is $L$ the same as the composite of the Galois conjugates $\sigma(K)$ for $\sigma\in G$? I know ...
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1answer
29 views

Degree 3 polynomial with coef in a field K: question on roots in a algebraic closure

I am asked the following question: Consider a field $K$ with characteristic different from 2 and 3, and the polynomial $f(t) = t^3 + pt + q \in K(t)$ with three distinct roots $\alpha_1, \alpha_2, \...
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1answer
55 views

If $[E:F]$ is finite and $\alpha \in E$ then there is an irr. polynomial in $F[x]$ with root $\alpha$

I'm studying for an exam and encountered a confusing proof of the following fact in my notes: Let $[E:F]$ be finite and $\alpha \in E$ then there is an irreducible polynomial $p(x) \in F[x]$ with $...
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3answers
69 views

For an irreducible polynomial $p$ and root $\alpha$, $[F[\alpha]:F]$ = degree of $p$

I'm studying for an exam, and I couldn't find the proof for the following theorem in my notes: For $p(x) \in F[x]$ irreducible and $\alpha$ a root of $p$ in some extension field, $[F[\alpha]:F] =$ ...
2
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1answer
251 views

Degree of extensions and their composite

Let $F<E$ and $F<K$ be finite extensions and assume that $EK$ is defined as composite of two fields. I need to show that $[EK:F] \leq [E:F][K:F]$, with equality if $[E:F]$ and $[K:F]$ are ...
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1answer
39 views

Integral domain and homomorphisms

This is a problem I've been working on. can you guys point out some uniqueness theorems you'd find helpful or an example proof? I'm pretty lost.
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76 views

Unique “splitting fields” for infinite polynomials

When creating the splitting field of a polynomial we can without loss of generality assume it has constant coefficient $1$ and splits as $\prod_{k=1}^n(1-z_kT)$. The splitting field is obtained by ...
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1answer
204 views

Are there nonisomorphic fields with isomorphic multiplicative groups?

This is false for finite fields as the multiplicative groups of finite fields are cyclic, and different cardinalities yield cyclic groups of different cardinalities. But I'm unsure how to proceed for ...
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3answers
447 views

Polynomials over finite fields

I’ve come across this problem in a coding theory course, and neither I nor several of my colleagues could solve it to our satisfaction. Let $F:=\mathrm{GF}\left(q\right)$ denote the field with $q$ ...
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59 views

The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ (...
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0answers
34 views

Fields and quotient ring

Let $P(X)\in{\mathbb{R}[X]}$ irreducible polynomial. Then $\mathbb{R}[X]/(P(X)=X^2+1)\approx{\mathbb{C}}$. If $P(X)=X^2+X+1$ also $\mathbb{R}[X]/(P(X))\approx{\mathbb{C}}$? Or for a arbitrary ...
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114 views

Algebraic number, Minimal polynomial and basis

Can someone help me with these question? a) Prove that $a = \sqrt{11} - \sqrt{2}$ is an algebraic number, by finding a polynomial with integer coefficients that has $a$ as a root. b) Find the ...
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1answer
76 views

Can I rotate divergence away? / Can get divergence from a rotation?

Let $v(x,y)$ be a two-dimensional vector field and let $R(x,y, \theta)$ be the two-dimensional rotation matrix which rotations a vector field around $(x,y)$ an angle $\theta$. The following two ...
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1answer
195 views

Should this be viewed as a serious issue with the meadow-theoretic approach?

Meadow theory (see here) allows us to apply the results and concepts of universal algebra to the study of fields. Obviously, this is very, very nice. However, I have the following issue with the ...
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1answer
133 views

Isomorphism of an extension field of a field of finite transcendence degree

The following is the proposition (1.4) of Mumford's book Algebraic Geometry If $\mathbb C$ has infinite transcendence degree over $k$, then every variety has a $k$-generic point. In the proof ...
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82 views

A field extension and equality up to isomorphisms

If we have in the category of rings two fields $F$ and $F'$ such that $F\hookrightarrow F'$ and $F'\hookrightarrow F$, do we have an isomorphism between $F$ and $F'$ ? If it is not always true, how ...
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0answers
286 views

Finite extensions of characteristic zero fields are simple.

I have been recently been introduced to field theory, and been told I should be able to understand a proof for this the result that: every finite extension of a field of characteristic zero is a ...
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1answer
522 views

Topological Quantum Field theories

I was wondering about the following on TQFTs. It is said that TQFTs have vanishing Hamiltonians $\hat{\mathcal{H}}$. Firstly, I would like to ask: Why is this so? Secondly, consider the ...
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2answers
77 views

How to prove that there exists a countable subfield of real numbers which is mapped into itself for any function f?

Given a function $f: \mathbb{R} \rightarrow \mathbb{R}$, how should I go about proving that there exists a countable subfield of $\mathbb{R}$, say $K$, which is mapped into itself? i.e., $f(K)\subset ...
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1answer
109 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to \...
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3answers
762 views

Computing basis of a field extension

Suppose we have an irreducible polynomial $p(x)$ of degree $n$ in $F[x]$, where $F$ is a field. Let $K$ be the field $F[x] / (p(x))$. We can consider the extension of $K$ over $F$ as a vector space ...
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1answer
497 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
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1answer
32 views

irreducibility of $x^2-a$ in $\mathbb{Z}_2[a]$

Let $a$ be transcendental over $\mathbb{Z}_2$ and let $F=\mathbb{Z}_2(a)$. Prove $p(x)=x^2-a$ is irreducible over $F$ I've been trying to understand this for a while now, but I'm having ...
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1answer
71 views

Showing a polynomial is irreducible over an extension field.

Show that the polynomial $$ x^3 - 3$$ Is irreducible over $$ Q(i, \sqrt2 ) $$ I'm a little stuck as I don't think I can use Eisenstein's criterion as we're not over the rationals. Also I know ...
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1answer
47 views

A counterexample about an inequality- Field extensions

Consider $A$ and $B$ two intermediate fields of the field extension F/K. I have already proved that $[AB:K]\leq {[A:K][B:K]\over [A\cap B: K]}$. I would like to find a simple example (for example, ...
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1answer
401 views

multiplicative group of infinite fields

I am stuck with the proof of below expression; "If F is an infinite field, then no infinite subgroup of F* (the multiplicative group of F) is cyclic." anyone can help?
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2answers
71 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
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1answer
113 views

Exercise about field extensions [duplicate]

Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...