# 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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### 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?
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### 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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### 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 ...
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### $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$

Let $p$ be a prime. Prove that $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2\bmod3$. So: If $p \equiv 2$ mod $3$, I have to show that every element of $\mathbb{F}_p[X]/(X^2+X+1)$ has ...
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### Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
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### Let $F$ be a field of order $2^n$. Prove that characteristic of $F$ is 2.

I figure that Lagrange's theorem and the fact that the characteristic of an integral domain is either $0$ or prime should be used, but just can't figure it out exactly.
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### How to show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$

So as title says I wanna show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ So I know that the minimal polynomial of $\mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ is $x^{4}-6x^{2}+7$ ...
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### Question about construction of an algebraic closure

$A=K[x]$,$\mathfrak{m}$ is a maximal ideal containing a principle ideal of $A$. every element of $K[x]/\mathfrak{m}$ can be described by form $f+\mathfrak{m}$. every element of $K$ is the ...
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### $\exists a \in \mathbb{F}_{11}$ such that $\mathbb{F}_{11}[x]/\langle x^5-a\rangle$ is a field

Prove that there exists an element $a \in \mathbb{F}_{11}$ such that the quotient ring $\mathbb{F}_{11}[x]/\langle x^5-a\rangle$ is a field. I wrote that it is equivalent to showing that there is an ...
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### Question about construction of an algebraic closure of a field

In Constructing algebraic closures by Keith Conrad, the author writes: Let $K$ be a field. We want to construct an algebraic closure of $K$, i.e., an algebraic extension of $K$ which is ...
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### Prove that $-1 \cdot x=-x$ [duplicate]

While working on a proof for class, I came to a point where I couldn't go any further without knowing that $-1 \cdot x=-x$. Is there a way to prove this using the axioms of a field?
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### {0} is unique maximal ideal when F is field [duplicate]

Let $R$ be a ring. Show that R is a field if and only if $\{ 0 \}$ is the unique maximal ideal of $R$. Thank you
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### Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
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### Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
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### $k$ algebraically closed field $\Rightarrow$ $V(f)\subset k^2$ infinite

Let $k$ be an algebraically closed field, and $f\in k[X,Y]$ a non-constant polynomial. Show that $V(f)\subset k^2$ is infinite. We solved this exercise in my tutorial class, but I have some questions ...
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### If $F\subseteq\mathrm{Mat}_n(\mathbb{Q})$, then $[F:\mathbb Q]\leq n$?

Let $F$ be a field contained in the ring of $n\times n$ matrices over $\mathbb Q$. Prove that $[F:\mathbb Q]\leq n$. I have an idea to consider a degree $n$ extension $K$ of $\mathbb Q$ and left ...
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### Composite of two simple extensions

Let $a, b$ be algebraic elements over a field $K$ and suppose at least one of these two elements is separable over $K$. Then, prove that there exists $c$ such that $K(a, b) = K(c)$.
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### Subfield of $\mathbb R$ with Algebraic closure as $\mathbb C$.

Does there exist a proper subfield of $\mathbb{R}$ whose algebraic closure is $\mathbb{C}$ ? A weaker question: Does there exist a proper subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is ...
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### Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
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### What are elementary field extensions?

While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I haven'...
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### Equivalent definition of purely inseparable field extension concerning extensions of morphisms.

Suppose $F$ is a field of characteristic $p$. I know there are many equivalent defintions that a field extension $K/F$ be purely inseparable, e.g., every separable element in $K$ over $F$ is in $F$, ...
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### Dimension of $K\subset L(\alpha)$ where $L$ is a field extension of $K$

Suppose $L$ is a field extension of $K$ and $\alpha$ an element in a field extension of $L$. Can we say $[K\colon L(\alpha)]=[K\colon K(\alpha)]$? I tried to prove this, but I couldn't come up with a ...
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### A polynomial with solvable Galois group and solution by radicals [duplicate]

Suppose $f(x)\in \mathbb{Q}[x]$ has a solvable Galois group, then we know that it can be solved in terms of radicals. But do we know how to explicitly write the solutions of $f(x)$ in terms of ...
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### What is an algebraic expression over a field structure?

I am working on a problem, and I am not understanding the language very well. Here is the setup of the problem: Consider the set $\{ 0, 1, 2 \}$ with the operations addition $(+)$ modulo $3$ and ...
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### Find all $\mathbb Q$-automorphisms of the field $\mathbb Q(\sqrt{5})$

My lecturer is off sick for now and the substitute is pretty bad at explaining this stuff. Note that when I write "$\mathbb Q$" I mean the symbol for rationals. I'm tasked with finding all $\mathbb Q$...
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I have a field extension $\mathbb Q (2^{1/3}) = a + b2^{1/3} + c2^{2/3}$ where $a,b,c\in \mathbb Q$. I want an elementary proof it indeed is a field. How to go about proving it contains its inverse $(\... 2answers 130 views ### Prove that$\left[\mathbb{Q}\left(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}}\right):\mathbb{Q}\right]=2^{n}$[duplicate] Let$p_{1},p_{2},\ldots,p_{n}$be$n$primes,$\left(p_{i},p_{j}\right)=1$if$i\neq j$. Prove that$\left[\mathbb{Q}\left(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}}\right):\mathbb{Q}\right]=...
Could anyone please throw some light on my questions? What is the splitting field of an $m$th-degree prime polynomial over $GF(q)$? Is it always $GF(q^m)$? How many $m$th-degree prime polynomials ...
### Is there a way to classify the finite fields where $x^2+1$ has a root?
It is a well-known theorem in number theory that $-1$ is a square in $\mathbb{F}_p$ if $p\equiv 1\pmod{4}$, and $-1$ is not a square if $p\equiv 3\pmod{4}$. Furthermore, $-1$ is easily seen to be a ...