# 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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### What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?

Let $Aut(\mathbb{C}/\mathbb{Q})$ be the field automorphisms of $\mathbb{C}$, and $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$ the subfield of $\mathbb{C}$ fixed by this group. I supsect that it is equal ...
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### Polynomial equations with finite field arithmetic

there are given 3 equations (they are connected with cyclic codes): $$s(x)=v(x)+q(x)g(x)$$ $$g(x)h(x)=x^7+1$$ $$s(x)=v(x)h(x)\bmod(x^7+1)$$ I have following data (for $GF(8)$ with generator ...
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### There are enough Galois extensions?

There are enough Galois extensions? For me enough means that every finite extension of a certain field is included in a Galois extension of that field, formally: "Given a field $k$, and a finite ...
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### Puiseux series over an algebraically closed field

Using the construction $R_n = K[t^\frac1n]$, $L_n = \text{Quot}(R_n)$ and $P = \bigcup_{n\in \mathbb{N}}L_N$ one automatically gets that the Puiseux series are a field. Nevertheless they are also an ...
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### Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?

I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
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### Galois Field Fourier Transform

there are two definitions for Reed-Solomon codes, as transmitting points and as BCH code ( http://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction ). On Wikipedia there is written that we ...
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### Binomial formula in $GF(2^m)$

there is a binomial formula: $$(x+y)^n=\displaystyle\sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$$ When operations are done in $GF(2^m)$ then all positive integers are reduced $\bmod2$, so binomial formula ...
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### About cyclic extensions of $\mathbb{Q}_p$

I'm trying to learn how to apply local class field theory and I thought about trying to enumerate some low degree abelian extensions of $\mathbb{Q}_p$. The easiest case is the quadratic extensions i.e....
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### Finitely generated field extensions

If $F=K(u_1,\ldots,u_n)$ is a finitely generated extension of $K$ and $M$ is an intermediate field, then $M$ is a finitely generated extension of $K$. I'm not exactly sure how to start this problem. ...
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### Relationship Between Field Automorphisms and Embeddings

I'm reading some Galois theory in Lang's Algebra, and he often refers to maps acting on elements of a field extension as embeddings in the algebraic closure of the base field (if I'm not mistaken). ...
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### Is an intersection of two splitting fields a splitting field?

Let $F$ be a field, and let $K_1$, $K_2$ be two splitting fields over $F$ (Suppose they are contained in a larger field $K$). Is $K_1\cap K_2$ necessarily a splitting field over $F$? The statement is ...
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### Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed?

If I have a field $K$ and an extension $L$ of $K$ such that all (non-constant) polynomials in $K[X]$ have a root in $L$, is the set of algebraic elements of $L$ over $K$ (the sub-field of all the ...
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### What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties. Perhaps this question could be put more generally as: ...
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### $\mathbb{Q}(\pi, i\pi)$ over $\mathbb{Q}$

Is $\mathbb Q(\pi,i\pi):\mathbb Q$ a simple extension?
### Set of elements in $K$ that are purely inseparable over $F$ is a subfield
Let $F\subset K$ be an algebraic field extension. Is the set of all elements of $K$ that are purely inseparable over $F$ necessarily a subfield of $K$?
Show that over any field $F$, the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors. Edited: This is my attempt: Let $f(x)=x^3-3x+1$. Let $a_1,a_2,a_3$ be the roots of $f$. ...