# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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, $[F[\alpha]:F] =$ ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?
### 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. ...
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 ...