# 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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### How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in GF(2^2)[X]$...
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### Finding the subfields of the cyclotomic field of order $5$

This is part of an exercise from Hungerford's Algebra: Find all intermediate fields in the field extension $F_5/\mathbb{Q}$, where $F_5$ is the cyclotomic extension of $\mathbb{Q}$ of order $5$. ...
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### Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff P|Q-S$ ...
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### Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes of ...
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### Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
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### Reducing splitting field

If we have splitting field: $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{2-i\sqrt{6}})$$ we can multiply these two zeroes and get $\sqrt{10}$ so we have $$L=\mathbb{Q}(\sqrt{2+i\sqrt{6}},\sqrt{10})$$ ...
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### List the elements of the field $K = \mathbb{Z}_2[x]/f(x)$ where $f(x)=x^5+x^4+1$ and is irreducible

Since $\dim_{\mathbb{Z}_2} K = \deg f(x)=5$, $K$ has $2^5=32$ elements. So constructing the field $K$, I get: \begin{array}{|c|c|c|} \hline \text{polynomial} & \text{power of $x$} & \text{...
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### Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
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### Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
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### What are the groups $\text{Hom}(F^\times\!, F^+\!)$ and $\text{Hom}(F^+\!, F^\times\!)$?

Background. Exercise 36 in Rose's A Course On Group Theory reads Prove that there is no field $F$ with $F^\times \cong F^+$. The problem is solved in characteristic $\ne 2$ by considering $-1$ ...
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### Proof that every field $F$ has an algebraic closure $\bar F$

I am reading the book A First Course in Abstract Algebra written by Fraleigh and I do not really understand the proof of theorem 31.22, that every field $F$ has and algebraic closure $\bar F$. I ...
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### How can we work with field extensions when our base fields aren't actually subfields?

I've been wondering this for a little while. Say we are working with the rational numbers $\mathbb{Q}$, and then we wish to talk about the extension fields $E$ of $\mathbb{Q}$, by which we mean the ...
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### Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
What do we mean when we say that a given prime $p$ splits completely in an algebraic extension of $\mathbb Q$? Are we talking about the splitting of prime ideals into unique factors? And, in that ...