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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Every subring of a field is a domain. Is this reciprocal?

I'm reading my notes on ring theory, and we proved on class that every subring of a field is a domain. Proof: Let $S \subseteq K$ be a subring of $K$, with $K$ a field. Let $x,y \in S$. If $xy=0$, ...
8
votes
1answer
148 views

Proving $f = x^8+x^7+x^3+x+1$ to be irreducible over $\mathbf{F}_2$.

I am taking all the irreducible polynomials over $\mathbf{F}_2$ of degree 1 to 4 inclusive and using the long division to determine whether the given $f$ is divisible by anyone of them and it is not ...
8
votes
2answers
200 views

Dimension of an algebraic closure as a vector space over its base field.

Let $k$ be an infinite field and $\bar{k}$ its algebraic closure. The Artin-Schreier Theorem tells us (among other things) that $[\bar{k}:k]=1,2,\infty$. There's a natural interpretation of ...
8
votes
1answer
67 views

Irreducibility of $x^{2n}+x^n+1$

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
8
votes
1answer
201 views

Intermediate field, normal closure and Galois group

Let $K/F$ be Galois with $G=Gal(K/F)$ and let $L$ be an intermediate field. Let $N\subseteq K$ be the normal closure of $L/F$. If $H=Gal(K/L)$ show that $Gal(K/N)=\cap_{\sigma\in G}\sigma ...
8
votes
1answer
104 views

Help understanding fields.

Hi guys I have a test this tuesday and I am given practice questions to do , and I have trouble understanding fields. Like I know by definition what they are, but applying them is kind of confusing. ...
8
votes
1answer
235 views

Generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$

I would like to find a generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$. This is a field since $x^{2}+3x+3$ is irreducible, so every coset with $bx+a\not=0$ as a ...
8
votes
1answer
80 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
8
votes
1answer
218 views

When does $\|z^2\|=\|z\|^2$

Let $k \in \mathbb{Z}$ and consider the field extension $K := \mathbb{Q}[\sqrt{k}]$. Define a norm on $K$ given by $\|p+q\sqrt{k}\| := \sqrt{p^2+q^2}$. For any $z \in K$, I was interested to know when ...
8
votes
1answer
193 views

Automorphisms of composite fields of finite extensions

If $\phi:\mathbb{Q}(a_1\ldots,a_n)\rightarrow\mathbb{Q}(b_1,\ldots,b_n)$ is an isomorphism of finite extensions of $\mathbb{Q}$ such that $\phi(a_i)=b_i$, can one extend $\phi$ to an automorphism ...
8
votes
1answer
673 views

A proof of Artin's linear independence of characters

I came up with a proof of Artin's linear independence of characters in field theory. The usual proof uses a clever trick devised by Artin. Since I'm not as clever as him, I prefer a proof which ...
8
votes
1answer
147 views

Should this be viewed as a serious issue with the meadow-theoretic approach?

Meadow theory (see here) allows us to apply the results and concepts of universal algebra to the study of fields. Obviously, this is very, very nice. However, I have the following issue with the ...
8
votes
1answer
389 views

Problem 18.7 in I. Martin Isaacs' Algebra

I am trying to solve the following problem in I. Martin Isaacs' Algebra: A graduate course, p.290: Let $f(X),g(X) \in F[X]$ and suppose $E \supseteq F$ is the splitting field both for $f(X)$ and ...
8
votes
1answer
739 views

Roots of unity and field extensions

Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the only roots of unity of $F$ are those in $K$, followed by an extension $L/F$ which is of the form ...
8
votes
0answers
80 views

Genus of $k(T)$?

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places $F$, and let $S$ be a nonempty finite subset of $X$. Then the genus of $F$ is equal to the ...
8
votes
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271 views

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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733 views

Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
7
votes
6answers
769 views

Are all finite fields isomorphic to $\mathbb{F}_p$?

I've recently started taking some algebra courses and I was wondering whether or not every finite field is isomorphic to $\mathbb{F}_p$, where $p$ is prime.
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votes
4answers
801 views

Show $x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$.

Show $p(x) = x^6 + 1.5x^5 + 3x - 4.5$ is irreducible in $\mathbb Q[x]$. By Gauss' Lemma, a primitive polynomial in $\mathbb Z[x]$ is irreducible in $\mathbb Q[x]$ if and only if it is irreducible in ...
7
votes
2answers
261 views

In field ($F, +, \cdot$) , how can I prove $x^2 =1\implies x=1,-1$

I'm a really confused about fields. I know that it means $x$ is the reciprocal element of itself, and I can easily show that $1^2=1$ (not as trivial for $(-1)^2$ though), but I'm not sure how it ...
7
votes
4answers
489 views

Proving that these two fields $\mathbb Z_{11}[x]/〈 x^2+1〉$ and $\mathbb Z_{11}[x]/〈 x^2+x+4〉$ are isomorphic with $121$ elements each.

I have been stuck in this problem for some time now. Prove that $x^2+2$ and $x^2+x+4$ are irreducible over $\mathbb{Z}_{11}$. Also, prove further $\mathbb Z_{11}[x]/\langle x^2+1\rangle$ and ...
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votes
2answers
94 views

Dimension of $\Bbb Q(e)$ over $\Bbb Q$?

The dimension of $\Bbb Q(\sqrt{2})$ over $\Bbb Q$ is finite since $\sqrt2$ is algebraic over $\Bbb Q$. But what about any transcendental number (say $e$)? Which is the smallest field containing $\Bbb ...
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1k views

Tensor product and compositum of fields

Let E/k, F/k be two arbitrary field extensions of k. My question is: Is there a field extension M/k s.t. E/k, F/k are subextensions of M/k? Alternatively, can we talk about compositum fields without ...
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votes
4answers
495 views

First examples in Galois theory

I'm studying Field Theory and after studying theorems and problems about extensions, splitting fields, etc... I'm starting with the first theorems of the Galois Theory itself. In order to see if I ...
7
votes
2answers
820 views

Show that an algebraically closed field must be infinite.

Show that an algebraically closed field must be infinite. Answer If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F ...
7
votes
4answers
248 views

Why is $\{a + b\sqrt2 + c\sqrt3 : a\in\Bbb{Z}, b, c \in\Bbb{Q}\}$ not closed under multiplication?

The set $R = \{a + b\sqrt{2} + c\sqrt{3}: a \in \Bbb{Z}, c, b \in \Bbb{Q}\}$ is not closed on multiplication, my textbook states. Why is this? And related to that: why then is $S = \{a + b\sqrt{2} : ...
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2answers
197 views

Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
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2k views

Every finite extension of a finite field is separable

I'm trying to prove that every finite extension of a finite field is separable. I found a solution on internet which says: Let $F$ be a finite field and $E$ be an extension of $F$ having $p^n$ ...
7
votes
1answer
685 views

Proving that a polynomial is not solvable by radicals.

I'm trying to prove that the following polynomial is not solvable by radicals: $$p(x) = x^5 - 4x + 2 $$ First, by Eisenstein is irreducible. (It is not difficult to see that this polynomial has ...
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2answers
1k views

Construct algebraic closure of $\mathbb{Q}$

In abstract algebra lecture, the lecturer wants to construct an algebraic closure of $\mathbb{Q}$. The construction is as follow: Suppose $\mathbb{Q_1}=\mathbb{Q}$. Let $\mathbb{Q}_2$ be the set ...
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votes
3answers
433 views

Do maximal proper subfields of the real numbers exist?

To clarify the problem, consider the field ${\Bbb R}$ as a field extension of ${\Bbb Q}$ using some sort of Hamel basis. Does there exist a field $F\subsetneq{\Bbb R}$ such that $F(\sqrt{2})={\Bbb R}$ ...
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votes
2answers
229 views

Explicit Galois theory computation in cyclotomic field

Let $p$ be an odd prime. Let $\zeta=e^{\frac{\pi i}{4 p}}$, thus $\zeta$ is a primitive $8p$-th root of unity. There is a unique $\mathbb Q$-automorphism $\tau$ of the number field ${\mathbb ...
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558 views

Determining subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ without Galois theory?

Somehow I had convinced myself recently that one could determine the subfields of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (here $\zeta_3 = \frac{-1+\sqrt{3}i}{2}$ is a primtive $3$rd root of $1$) without ...
7
votes
1answer
373 views

For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$

Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that differs from $F$. Why is $x$ algebraic over $K$? Thanks a lot!
7
votes
4answers
208 views

Fields of arbitrary cardinality

So I took an introductory abstract algebra course a few semesters ago, and we were shown that groups and rings can both be made into products, i.e. if I have some group $G$ (resp. ring $R$) and some ...
7
votes
1answer
127 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
7
votes
1answer
158 views

Extension degree of residue field.

Let $k$ be a field, and $A$ be a finitely generated $k$-algebra with $\text{dim}(A)\leq 1$. Then for any maximal ideal $\mathfrak{m}$ of $A$, does this inequality $[A/\mathfrak{m}:k]<\infty$ hold? ...
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votes
3answers
514 views

Subextension of a finitely generated extension of fields

If $E/K$ is a finitely generated field extension and $F$ is an intermediate field how can I prove that $F/K$ is finitely generated?
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2answers
554 views

Derivations in a ring. What applications do they have outside algebra?

INTRODUCTION Let $R$ be any ring, possibly without unity. We call a function $d:R\longrightarrow R$ a derivation on $R$ if it satisfies the following conditions. $(1)$ It is an endomorphism of the ...
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222 views

Field Extensions

Let $L/K$ a finite extension and $f(x)\in K[x]$ a non-linear irreducible polynomial. Prove that if $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ then $f(x)$ has no roots in ...
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232 views

A shortcut in Galois theory

How could we prove Galois correspondence without using Dedekind’s Lemma on group characters, Artin’s lemma and the primitive element theorem ? I just came across Meinolf Geck's article On the ...
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348 views

Why are the surreals considered “recreational” mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, ...
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209 views

Proving that $X^6-15X^4-6X^3+75X^2-90X-116$ is irreducible over $\mathbb Q$

When asked to find the minimal polynomial of $\sqrt[3]{3}+\sqrt[2]{5}$ over $ \mathbb Q$, I easily found out that $X^6-15X^4-6X^3+75X^2-90X-116$ has $\sqrt[3]{3}+\sqrt[2]{5}$ as a root. It's very ...
7
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1answer
145 views

A first order theory whose finite models are exactly the $\Bbb F_p$

Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of ...
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132 views

Equivalent definition of algebraically closed

In Hungerford's Algebra text, it is stated that a field $K$ is algebraically closed iff there exists a subfield $F$ such that $K$ is algebraic over $F$ and all polynomials in $F[x]$ split in $K[x]$. ...
7
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154 views

prove that the field extension is cyclic

Let's define the sequence $x_0=0$ and $ x_{i+1} = \sqrt{x_i+2}$ taking always the positive root. Prove that the field extension $\Bbb Q \subset \Bbb Q(x_i) $ is cyclic with degree $2^i$ Well.. at ...
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2answers
205 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...
7
votes
1answer
207 views

Cyclotomic polynomials and Galois group

Let $\zeta\in \mathbb C$ be a primitive $7^{th}$ root of unity. Show that there exists a $\sigma\in \operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ such that $\sigma(\zeta)=\zeta^3$. I already know ...
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2answers
103 views

Euclid and finite fields

In 300 BC or so Euclid pointed out that if $S$ is any finite set of prime numbers then the prime factors of $1+\prod S$ are not in $S$, so that $S$ can always be extended to a larger finite set. Much ...
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votes
2answers
201 views

If $E/F$ is algebraic and every $f\in F[X]$ has a root in $E$, why is $E$ algebraically closed?

Suppose $E/F$ is an algebraic extension, where every polynomial over $F$ has a root in $E$. It's not clear to me why $E$ is actually algebraically closed. I attempted the following, but I don't think ...