Questions tagged [field-theory]

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. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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Show that $\ I=(x^n)$ for some $n$, an element of $\Bbb N\cup\{0\}$. [on hold]

Assume $k$ is a field, $k[[x]]$ is the ring of formal power series. Let $I$ be an ideal of $k[[x]]$. Show that $\ I=(x^n)$ for some n, an element of $\Bbb N\cup\{0\}$.
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Algebraic independence over subfields and algebraic closure?

Are my following conjectures formulated linguistically and mathematically correct? 1.) $K$ be a field, $O$ a superfield of $K$, and $v_1,...,v_n$ be algebraically dependent over $K$. $v_1,...,v_2$ ...
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Finite extension of the inclusion $\mathbb F_p \to \Omega$ is injective?

Let $\Omega$ be an algebraically closed field of characteristic $p$ and $\mathbb F_p \to \Omega$ be the standard inclusion map. I wonder if for any finite extension $K/{\mathbb F_p}$, the extension of ...
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sub-resultant GCD algorithm for polynomials over a field

I'm trying to implement the algorithm for factoring polynomials over number fields on page 145 of Henri Cohen's A Course on Computational Algebraic Number Theory, but I'm not sure how the application ...
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Conjecture about extending a field

My conjecture is motivated by the desire, given a field $F$, to find an extension field by "adjoining" elements. Let's say $\forall x\in F[x*x+1\neq0]$ and we want to "adjoin" an element $α$ ...
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Minimal polynomial of $\sqrt{2+\sqrt[3]{3}}$ over $\mathbb{Q}$

About 2 weeks ago, I tried to solve the following problem. Find the minimal polynomial of $\alpha=\sqrt{2+\sqrt[3]{3}}$ over $\mathbb{Q}$. My attempt First, I tried to find the polynomial with ...
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Computing $[\mathbb{F}_p(x):\mathbb{F}_p(x)^p]$

If $F=\mathbb{F}_p(x)$ is the rational function field in one variable over $\mathbb{F}_p$, find $[F:F^p]$.* Here's where I am at with this question: Using the multinomial theorem (and some tears), I ...
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Minimal polynomial over a field extension

I came across this question while doing some exercises at the end of the chapter. I would like someone to comment on my solution (on its correctness, completeness and approach): Let $\alpha\in E$, ...
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Number fields with only trivial field automorphism

Fields with trivial automorphism group have been addressed in a question on Mathoverflow (see this). However, I didn't find sufficient information of number fields with such properties. Q. Let $K$ ...
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Amann/Escher, Analysis I, Exercise I.11.8: field automorphisms of $\mathbb{C}$ which leave the elements of $\mathbb{R}$ fixed

I'm doing Exercise I.11.8 from textbook Analysis I by Amann/Escher. Show that the identity function and $z \mapsto \overline{z}$ are the only field automorphisms of $\mathbb{C}$ which leave the ...
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Some confusion about Gaussian ring [closed]

Is $\mathbb{Z} [i]$ is field ? yes/No yes, I thinks it will field because it is integral domain Is its True ?
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How is $E = S(x)$, where $E = k(x), k$ a field, and S= k(I) of all rationals functions of $I = I(x) = \frac{(x^2 -x+1)^3}{x^2(x-1)^2}$?

In Section G, Part II of Emil Artin's Galois Theory, in the first example.of the section, the author says : Here, $k$ is a field, $E=k(x)$ is the space of all rational functions of variable $x$. We ...
Why $\frac{R[x]}{<x^2-1>}$ is not a field, but if $Q[x]$ is there instead of $R[x]$ it is. How to check $<x^2-1>$ is maximal ideal or not in the easiest way possible?