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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Irreducible cyclotomic polynomial

I want to know if there is a way to decide if a cyclotomic polynomial is irreducible over a field $\mathbb{F}_q$?
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1answer
124 views

Algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$

Is there a concrete description of the algebraic closure of $(\mathbb{Z}/p\mathbb{Z})(T)$?
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5answers
463 views

why are subextensions of Galois extensions also Galois?

An algebraic extension of fields $L|K$ is defined to be a Galois extension iff the set of elements of $L$ invariant under the action of $Aut_K L$ is $K$. Apparently in the sequence of field ...
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votes
1answer
1k views

Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in ...
8
votes
1answer
3k views

Existence of irreducible polynomials over finite field

Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$? One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge ...
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2answers
1k views

Problem in Jacobson's Basic Algebra (Vol. I)

It is section $4.4$, exercise number $5$. It says the following: Let $F$ be a field of characteristic $p$. Show that $f(x)=x^p-x-a$ has no multiple roots and that $f(x)$ is irreducible in $F[x]$ if ...
8
votes
2answers
519 views

Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.

In an answer to another question I used the fact that $\mathbb{Q}(\zeta_m)\subseteq \mathbb{Q}(\zeta_n)$ if and only if $m$ divides $n$ (here $\zeta_n$ stands for a primitive $n$th root of unity, ...
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votes
2answers
2k views

Degree of $\sqrt{2}+\sqrt[3]{5}$ over $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt[3]{5})$

I'm self-studying field extensions. I ran over an exercise which I can't completely solve. (I haven't yet started studying Galois theory, and I think this exercise isn't meant to be solved using it, ...
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2answers
162 views

The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
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1answer
1k views

Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is ...
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votes
2answers
257 views

On irreducible factors of $x^{2^n}+x+1$ in $\mathbb Z_2[x]$

Prove that each irreducible factor of $f(x)=x^{2^n}+x+1$ in $\mathbb Z_2[x]$ has degree $k$, where $k\mid 2n$. Edit. I know I should somehow relate the question to an extension of $\mathbb Z_2$ of ...
8
votes
1answer
221 views

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
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votes
2answers
1k views

A question regarding normal field extensions and Galois groups

The following is possibly true but I can't find a corresponding theorem: If $E/F$ is the splitting field of some polynomial in $F$ and $F \subset K \subset E$ then: $Gal(E/K)$ normal subgroup of ...
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votes
1answer
113 views

Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
8
votes
1answer
389 views

Cyclotomic extensions of $\Bbb Q$

Let $n>4$, and $(h,n) = 1$. How to show that $[\mathbb{Q}(\tan 2 \pi h/n):\mathbb{Q}]= \phi(n)$ or $\phi(n)/2$ or $\phi(n)/4$ respectively if $\gcd(n,8)<4$ or $\gcd(n,8)=4$ or ...
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votes
1answer
569 views

Why $E$ is the algebraic closure of $K$?

Let $E/K$ be a separable, algebraic extension such that every noncostant polynomial in $K[x]$ has a root in $E$, then $E$ is an algebraic closure of $K$. Could you help me to solve this exercise? ...
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votes
3answers
210 views

How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$?

Let $p>2$ be an odd prime. Let $\mathbb F_{p^n}$ be the field with $p^n$ elements. How many solutions of the equation $x^2-y^2=1$ are there with $x,y\in\mathbb F_{p^n}$? My work: Char $F=p$. ...
8
votes
1answer
213 views

Can we always find a primitive element that is a square?

Let $L/\mathbb Q$ be a galois extension. The Primitive element theorem says, that there is an element $\alpha \in L$, so that $L=\mathbb Q(\alpha)$. Can I always find an element $\beta \in L$, so ...
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votes
2answers
183 views

Intermediate field between $F$ and $F(x)$

Suppose that $F$ is a field and that $u \in F(x):= \{PQ^{-1}:P,Q \in F[x], Q\neq 0 \}$, so that $F \subseteq F(u) \subseteq F(x)$. Is there a general method for determining $[F(x):F(u)]$? For my ...
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votes
2answers
406 views

Is a completion of an algebraically closed field with respect to a norm also algebraically closed?

Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
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3answers
2k views

Example of a complete, non-archimedean ordered field

I'm looking for a concrete example of a complete (in the sense that all Cauchy sequences converge) but non-archimedean ordered field, to see that these two properties are independent (an example of ...
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1answer
95 views

A field which is not algebraically closed but has no extensions of a fixed degree(s)?

Consider the field $k$ obtained as the union of all finite towers of degree $2$ extensions over the rationals. Then $k$ has no degree $2$ extensions, yet $k$ admits extensions of every other finite ...
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votes
2answers
90 views

Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
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votes
1answer
78 views

On the existence of field morphisms

Let $K$ and $L$ be two fields, does the existence of two field morphisms $f\colon K\rightarrow L,\ g\colon L\rightarrow K$ imply that, as abstract fields, $K\cong L$ (not necessarily via $f$ or $g$)?
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votes
2answers
136 views

Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. ...
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votes
1answer
184 views

Question about the Galois extension of a given field extension

Let $K=\mathbb{Q}(\omega)$ be given, where $\omega^3=1$. I want to know: (1) Whether there is a Galois extension $L/\mathbb{Q}$ containing $K$ such that $\mathrm{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_4$? ...
8
votes
1answer
122 views

If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$

Suppose $K$ is a field and $F$ is an algebraic extension of some degree $n=(F:K)$. It is stated that the field of rational functions $F(x)$ is in fact an algebraic extension of the field $K(x)$ and ...
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1answer
77 views

Are all quintic polynomials of this type not solvable by radicals?

The author of my textbook argues that the quintic polynomial $3x^5-15x+5$ is not solvable by radicals over $\mathbb{Q}$ by showing that the Galois group of $3x^5-15x+5$ over $\mathbb{Q}$ is isomorphic ...
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1answer
68 views

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
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1answer
154 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
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1answer
278 views

How to show that $(\mathbb{Q}(\sqrt{2}))^{\times} \cong (\mathbb{Q}(\sqrt{3}))^{\times}$

How to show that $$(\mathbb{Q}(\sqrt{2}))^{\times} \cong (\mathbb{Q}(\sqrt{3}))^{\times}$$ where $(\mathbb{Q}(\sqrt{2}))^{\times}$ is multiplicative group of $\mathbb{Q}(\sqrt{2})$. Mapping ...
8
votes
2answers
213 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 ...
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1answer
89 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
75 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
214 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
260 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
784 views

$K/E$, $E/F$ are separable field extensions $\implies$ $K/F $ is separable

I wish to prove that if $K/E$, $E/F$ are algebraic separable field extensions then $K/F$ is separable. I tried taking $a\in K$ and said that if $a\in E$ it is clear, otherwise I looked at the minimal ...
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1answer
219 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 ...
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1answer
203 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 ...
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1answer
765 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
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1answer
176 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 ...
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1answer
431 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 ...
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0answers
88 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 ...
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299 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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0answers
505 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
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6answers
1k 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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4answers
855 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
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2answers
268 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
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4answers
524 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 ...
7
votes
2answers
101 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 ...