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 ...

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Hilbert's Nullstellensatz - question about generalization

It is well-known that if $k$ is algebraically closed field that has infinite transcendence degree over the prime field $\mathbb{Q}$ or $\mathbb{F}_p$ then the maximal ideals of $k[x_1,...,x_n]$ are of ...
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What is the smallest $m>0$ for the Frobenius automorhism of a Galois Field to be the identity?

Surprisingly scarce information on this particular problem. $\phi$ is the Frobenius automorphism of $GF(p^n)$ for some prime $p$. Find the smallest $m>0$ such that $\phi^m$ is the identity ...
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Finite Groups as Galois Groups of Field Extensions [duplicate]

Let $F$ be a field, and $x_{1}, \dots, x_{n}$ free variables. Let $K=F(x_{1},x_{2},\dots,x_{n})$, being the fraction field of the ring of polynomials in the variables $x_{1},\dots,x_{n}$. Show that ...
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Field extensions of $\mathbb{Q}$, $\mathbb{Q}(\xi_7)$ and $\mathbb{Q}(\xi_7+\xi_7^{-1})$

Let $\xi_7$ denote the complex number $e^{2\pi i/7}$ and let $\beta = \xi_7+\xi_7^{-1}$, consider the field extensions $\mathbb{Q} \subset \mathbb{Q}(\beta) \subset \mathbb{Q}(\xi_7)$. Determine ...
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Explicit way of finding particular field elements

Suppose I am working with the field $\mathbb{F}_{5^2}$ and I know that there exists a primitive element $\gamma$ that satisfies the following equation $\gamma^2 = \gamma + 3$. I want to calculate ...
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Dimension of the algebraic closure of a continuum field of characteristic zero

Let me start by saying that I have no idea in algebra/number theory/whatever, so, please, forgive my ignorance. Let $\mathbb{F}$ be a field of characteristic zero and continuum cardinality, which ...
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Proving existence of an element of trace 1

Let $F=\mathbb{F}_{q}$ be a finite field of order $q=2^{n}$ and let $\beta$ be a primitive element of $F$. I would like to prove that if $q>4$, then for each $1\leq i \leq \frac{q-2}{2}$, there ...
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Prove $F=K(u)$ if $F$ is a cyclic extension of $K$. (Hungerford, exercise V.7.5) [closed]

Prove that if $F$ is a cyclic extension of $K$ of degree $p^n$ ($p$ prime) and $L$ is an intermediate field such that $F=L(u)$ and $L$ is cyclic over $K$ of degree $p^{n-1}$, then $F=K(u)$
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How many different field homomorphisms can we have between any two given fields?

How many field homomorphisms can we have between any two given fields? That is, if $A$ and $B$ are fields, what can we say about the cardinality of $\text{Hom}(A,B)$? I don't know much abstract ...
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Action on its generators of splitting field of $x^4 +5$

Splitting field is $K=\mathbb Q (\sqrt[4]{-5}, i)$ Degree of $K$ over rationals is $8$ so the galois group $G=\text{G}(K/ \mathbb Q)$ has order $8$. $x^4 +5$ is irreducible so there is one orbit ...