# 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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### Solvability and reducibility of a polynomial in a “chain” of finite fields

This question is generalized based on my previous question: Is $x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$? Problem: Consider an irreducible polynomial $f = x^4 + x^3 + 1$ in ...
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### What is $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p(X)} \mathbb{F}_p(\sqrt{X})$?

I am trying to understand what $\mathbb{F}_p(\sqrt{X})\otimes_{\mathbb{F}_p (X)} \mathbb{F}_p(\sqrt{X})$ is. $\mathbb{F}_p(\sqrt{X})$ is the field of rational functions in $\sqrt{X}$. What is it ...
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### does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
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### Is there an infinite field F with char(F)=p and not algebraically closed field?

Is there an infinite field F with characteristic of the field $F$ is $p$ (p is prime) and not algebraically closed field ?
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### Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
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### Is $x^{2\cdot 3^n}+x^{3^n}+1$ irreducible (mod 2)?

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, it seems that the polynomials of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$ are irreducible. ...
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### What does a finite extension look like?

Let $F$ be a field. Let $\Gamma$ be an indexing set. Let $\lbrace \theta_i\rbrace_{i\in\Gamma}$ be a collection of of elements of $\overline{F}$. Let $L=F(\theta_i\text{ }|\text{ }i\in\Gamma)$ (the ...
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### Proof: Zero is less than one

How can one show/prove that $0<1$? $1$ is not actually defined to be greater than zero, but I think it can be proven. I already know that the real numbers are an ordered field and I am familiar ...
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### $F(X)$ as a subfield of $F((X))$ of formal Laurent series

$F(X)$ is a subfield of $F((X))$ by considering the Laurent expansion of rational functions at the origin. So what is indeed the degree of this field extension $F((X))/F(X)$? Or this is an infinite ...
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### Why is analysis over the complex numbers so useful vs say other fields?

First I'll state a statement that I hope is false, but I do not know if it is: "Complex analysis is used a lot compared to analysis over other fields (as in it gives a lot of results like the prime ...
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### A question concerning non-algebraic extension.

Let $\tau:F \to \overline{F}$ be a field embedding. Then is $\overline{F}/\tau(F)$ algebraic extension? I don't think so but I cannot find a counterexample. Would you let me know a counterexample?
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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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### prove that homomorphism (rings) from a field to ring is bijective or the zero homomorphism.

$F$ is a field and $R$ is a ring.$\:\phi :F\rightarrow R$ is a ring homomorphism. I need to prove that it is bijective or it is $\phi =0$. I tried to use some how the fact that I have opposites in F, ...
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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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### Extending isomorphism to compositum of fields

Let $F$ be a field and $\Gamma$ be an indexing set (possibly infinite). Let $K$ be another field. There is an isomorphism $\sigma:F\longrightarrow K$. Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a ...
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### For an algebraically closed field $k$, an ideal $I$ of $k[x]$ is maximal if and only if $I = (x-c)$

This is an exercise $4.21$ on a page $155$ from a textbook "Algebra: Chapter $0$" by P.Aluffi. Let $k$ be an algebraically cloased field, and let $I \subseteq k[x]$ be an ideal. Prove that $I$ is ...
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### Isomorphism of fields

In set-theory, one of the standard result (Bernstein's theorem) is that if there is an injection from $A$ to $B$ and an injection from $B$ to $A$, then there is a bijection from $A$ to $B$. Consider ...
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### How to determine redundant elliptic curves?

When we enumerate elliptic curves $y^2 = x^3 + ax + b$ over a finite field, how do we determine redundant ones, i.e. ones that are equivalent to others?
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### Compositum of an infinite family of fields

Let $F$ be a field and $\Gamma$ be an indexing set (possibly infinite). Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a collection of subfields of $\overline{F}$. Let $E\subseteq \overline{F}$ be the ...
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### is the compositum of a family of algebraic extensions algebraic?

Let $F$ be a field contained inside another field $K$. Let $\Gamma$ be an indexing set (possibly infinite). Let $\lbrace E_i\rbrace_{i\in\Gamma}$ be a collection of algebraic extensions of $F$ ...
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### Why consider ramification only over number fields?

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.
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### Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
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### Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$

Find the Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$, for $\zeta_{3}$ being a third primitive root of unity. It's easy to show this is a Galois extension since it will be ...
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We take $\mathbb{Q}_p$ to be the completion of $\mathbb{Q}$ with respect to $|\cdot|_p$. If $x=\sum_{j=k}^{\infty} a_jp^j$ is some element in $\mathbb{Q}_p$, then how exactly does $|\cdot|_p$ extend? ...
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### Points on a p-adic circle

I was wondering if anybody could point me to any interesting geometry (if there is any) that the p-adic circle has. Specifically, let $G_p=\{ (x,y)\in (\mathbb{Q}_p)^2 \,:\, x^2+y^2=1\}.$ Does $G_p$ ...
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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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### $i^i$ 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 ...