# 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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### Does there exist any isomorphism between $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 3]$? [duplicate]

Does there exist any isomorphism between the fields $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 3]$ ?
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### Is it true that $\mathbb{Q}(\sqrt{2},e^{2i\pi/3}) = \mathbb{Q}(\sqrt{2}+e^{2i\pi/3})$?

Is it true that $\mathbb{Q}(\sqrt{2},e^{2i\pi/3}) = \mathbb{Q}(\sqrt{2}+e^{2i\pi/3})$? I know that $[\mathbb{Q}(\sqrt{2},e^{2i\pi/3}):\mathbb{Q}]=2\times2=4$. By using WolframAlpha (cheating), I know ...
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### Show that $n$ is a divisor of $[L:K]$

Let $L/K$ be a field extension. I want to show that if the extension $L/K$ is finite and $a\in L$ has a minimal polynomial of degree $n$, then $n$ is a divisor of $[L:K]$.  I have done the ...
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### Show that $[L:K]=1 \Leftrightarrow L=K$

Let $L/K$ be a field extension. I want to show that $$[L:K]=1 \Leftrightarrow L=K$$  I have done the following: For the direction $\Rightarrow \ :$ Since $[L:K]=1=\text{dim}_KL$ we ...
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### Determining the minimal polynomial of $\omega := e^{2πi/p}$ over $\mathbb{Q}[\omega + \omega^{-1}]$

Let $p ≠ 2$ be a prime number, and $\omega = e^{2πi/p}$. I now want to find the minimal polynomial of $\omega$ over the field $\mathbb{Q}[\omega + \omega^{-1}]$. I must admit that I don't really know ...
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I'm studying Field and Galois Theory with different books and now I have a doubt about what is the exact statement of Galois' theorem. Some books define radical and solvable extensions but other books ...
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### Cyclotomic polynomials being irreducible over Q

So, task is to, using algebra, write polynomial $X^n-1$ as a product of irreducible polynomials over Q. Our prof told us that the solution is : $X^n-1 = \prod_{d|n} \Phi_d(x)$ where $\Phi_d(x)$ is d-...
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### Why working in compact spaces?

I am trying to study moduli spaces of stable curves with n-marked points, $M_{0,n}$. However, in general the texts generally talk about the closure of this space, $\overline{M_{0,n}}$. My question, ...
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### In abstract algebra, what is an intuitive explanation for a field?

Wikipedia has the following to say about fields. In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or ...
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### $i^i$ is real and also infinitely many sets in $\mathbb C - \mathbb R$

$i^i$ is real and also infinitely many sets in $\mathbb C - \mathbb R$ where operations in proper superset/field maps to a proper subfield. Is this of mapping between superfields to subfields of any ...
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### Fields that are vector-spaces over the set of real numbers [duplicate]

Can somebody enlighten me on how to prove that there exists no field that's also a vector space over the real numbers of dimension greater than 2?
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### Orders of elements in multiplicative groups of fields with positive characteristic

Suppose that $k$ is a field with positive characteristic $p$. I want to show that for each $x\in k$ the set $\{x^n\colon n\in \mathbb{N}\}$ is finite. My intuition tells me that I should use Fermat's ...
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### Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
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### Degree of the difference of two roots

Let $f\in \mathbb Q[x]$ be an irreducible monic polynomial of degree $n$ and let $\alpha,\beta\in \overline{\mathbb Q}$ be two distinct roots of $f$. Is it possible to find a lower bound on the degree ...
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### Example of a communtative ring with two operations where the identity elements are not distinct?

I was introduced to the definition of a field today, as a communtative ring with two operations, like $\Bbb{R} = \langle R, +, -, \cdot, ^{-1} \rangle$ and all the usual axioms; commutativity, ...
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### Degree of a field extension for three fields

Given field extension $L/K$ and fields $E_1,E_2$ with $$(1)\ K\subset E_1\subset L,\ [E_1:K]=n_1$$ $$(2)\ K\subset E_2\subset L,\ [E_2:K]=n_2.$$ If $\gcd(n_1,n_2)=1$ then $K=E_1\cap E_2$. Proof: ...
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### Workability of linear equation solving methods for different fields?

So far, I have mostly done linear algebra over $\mathbb R$ and $\mathbb{C}$. I know there exist very many methods to solve equation systems for those fields, of which a few are Gaussian Elimination, ...
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### A field in which every element (that is not 1 or 0) is a root of -1

Let $\mathbb{F}$ be a field with $char(\mathbb{F}) \neq 2$ such that for every element $q \in \mathbb{F}$ if $q \neq 0$ and $q \neq 1$ then there is a power n such that $q^n = -1$. (E.g. $\mathbb{F}_3$...
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### Linking two theorems: on algebraic closure and on minimal splitting field

Consider following two statements from same book of Cohn: Basic Algebra. Proposition 7.3.2: If $k$ is any field and $\mathcal{F}$ is a set of polynomials over $k$, then any two minimal splitting ...
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### Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
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### Let $K$be a field and $f \in K[x]$. Then there exists a splitting field for $f$ over $K$

Let $K$ be a field and $f \in K[X]$. Then there exists a splitting field for $f$ over $K$. I don't understand what this means, I think I am interpreting it wrongly. Take $x^2+1 \in \Bbb{Q}[X]$ then ...
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### Polynomial with infinite roots

I started Ring theory recently and I came across this statement while reading polynomial rings.. If $F$ is an infinite field and let $f(x)\in F[x]$ . If $f(a)=0$ for infinitely many elements $a$ ...
I am wondering whether the following statement is true or not? Given any field $F$, there exists a proper field extension $K$ of $F$.
### Construction of field extension for $[E:\mathbb F_{11}]=3$
Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1$ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...