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 ...

learn more… | top users | synonyms (1)

6
votes
4answers
201 views

Show that over any field $K$, such that $\mathbb Q \subset K$ the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors

Show that over any field $K$, such that $\mathbb Q \subset K$ the polynomial $x^3-3x+1$ is either irreducible or splits into linear factors I think if we prove that: if $K$ contain one root it ...
2
votes
3answers
155 views

$\mathbb{R}\subset\mathbb{F}\subset\mathbb{C}$ implies $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$?

Let $\mathbb{F}$ a field such that $\mathbb{R}\subset\mathbb{F}\subset\mathbb{C}$. Prove that $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$. What I know is, that if ...
3
votes
2answers
279 views

Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b $ has degree of extension 3

Find conditions on $a$ and $b$ such that the splitting field of $x^3 +ax+b \in \mathbb Q[x]$ has degree of extension 3 over $\mathbb Q$. I'm trying solve do this question, it seems very difficult to ...
0
votes
2answers
418 views

Proving that this field extension is normal

Let's consider the polynomial $f=x^6+3 \in \Bbb Q[x]$. I have to prove that for some root $\beta $ of $f$. The extension $ \Bbb Q (\beta) $ is galois. In other words if $ {\root 6 \of { - 3} } $ ...
1
vote
1answer
165 views

Polynomial of degree $n$ over a field of characteristic $p>0$ has at most $n/p$ distinct roots

Let $f$ be a polynomial of degree $n$ over a field $F$ of characteristic $p$. Suppose $f'=0$. Show that $p\mid n$ and that $f$ has at most $n/p$ distinct roots. I can't solve this question, any help ...
0
votes
0answers
84 views

Algebraic extension of a complete field is complete

Can someone please give me a reference(or a sketch of a proof), where I can find a proof that any algebraic extension of a complete field is complete ?
2
votes
1answer
477 views

The field trace is surjective over the fixed field

This problem it's from Stewart Galois Theory book. I want to solve it only using the theorems used on the book. Or at least simple tools. Let $K$ be a field of characteristic $0$. And let $L/K$ be a ...
1
vote
1answer
97 views

Abstract Algebra elementary question

Let $F = \mathbb{Q}(\pi^3)$. How can we find a basis for $F(\pi)$ over $F$ ?
7
votes
2answers
1k views

Why are fields with characteristic 2 so pathological?

For example, over fields with characteristic 2, there exist nonzero symmetric nilpotent matrices, and nonzero matrices could be simultaneously symmetric and anti-symmetric. I wonder why characteristic ...
1
vote
2answers
166 views

Find the smallest normal extension

Find the smallest normal extension (up to isomorphism) of $\mathbb Q(\sqrt[4]2)$ in $\overline {\mathbb Q}$ (the algebraic closure of $\mathbb Q)$ My atempt: $\mathbb Q(\sqrt[4]{2},i)$ is a normal ...
6
votes
1answer
548 views

Normal Extension Equivalent to Same Degree Irreducible Factors (Hungerford, Exercise V.3.24)

I am doing exercises from Hungerford's Algebra for preparation of my exam. I would appreciate some help in a part of the proof of the following question: Exercise V.3.24. An algebraic extension ...
3
votes
3answers
656 views

The splitting field of $x^3+x^2+1$ over ${\Bbb Z}/(2)$

Let $F=\mathbb Z/(2)$. The splitting field of $x^3+x^2+1\in F[x]$ is a finite field with eight elements. my attempt of solution: If $\alpha$ is a root in this polynomial in its splitting field, then ...
2
votes
2answers
449 views

How to prove that for every finite field its cardinality is $p^n$?

How to prove that for every finite field its cardinality is $p^n$ where $p$ is prime and $n\in\mathbb{N}$? Thanks in advance!
1
vote
3answers
98 views

Making a set into a Field

Let $K$ be the set of the following four-tuples of elements of $GF(3)$: $$K = \{(0,0,0,0),(1,2,1,1),(2,1,2,2),(1,0,0,1),(2,2,1,2),(2,0,0,2),(0,1,2,0),(0,2,1,0),(1,1,2,1)\}.$$ Define operations ...
0
votes
1answer
20 views

Verifying the structure of a field

Let F be a field and $G=F\times F$ Define addition by $(a,b)+(c,d)=(a+c,b+d)$ and multiplication by $(a,b)\cdot(c,d)=(ac,bd)$ Does these operations define a field on G? I'm fairly comfortable with ...
3
votes
1answer
128 views

What can be said about the Galois group of $f(g(x))$?

Supposing we know the Galois groups of $f(x)$ and $g(x)$ over $K$, what can be said about the Galois group of $f(g(x))$? I suppose we can restrict the question to normal polynomials over ...
5
votes
2answers
311 views

How do I prove that this polynomial is irreducible?

How do I prove that $x^4+1$ is an irreducible polynomial over $\mathbb Q$? I've already tried the Eisenstein criterion which gives to me any results to solve this question, I need help here. Thanks
1
vote
1answer
83 views

Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ with the following property?

Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ (proper subset) as follows: $\mathbb{R}$ is a vector space over $K$ and has no finite generating set and $K$ is a vector ...
0
votes
3answers
94 views

Technique for showing an element is not in a field?

I have an extension $\mathbb{Q}(5^{1/4}, i)$, and I want to show that $4^{1/4}$ is not contained in it. (I hope what I am trying to prove is true!) Anyways, my natural starting point is to assume ...
1
vote
1answer
165 views

Equality of two sigma algebras?

Assume that $F_1$ and $F_2$ are two independent sigma fields. We know that union of $F_1$ and $F_2$ is not necessarily a sigma-field. Suppose we define $ \mathcal{A} = \{A \cap B: A\in F_1, B\in F_2\} ...
3
votes
1answer
1k views

Sigma field generated by a random variable!

Consider $A = \text{the sets of the form {X $\le$ x}}$. The goal is to prove that $\sigma(A) = \sigma(X)$. The question seems obvious to me but I just don't know how to prove it. I also have ...
9
votes
2answers
327 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
1
vote
1answer
151 views

Algebraic Property inherited by a “lift”?

Suppose $K$ is a field, and $K\subset E$ is an algebraic extension, and $K\subset F$ is any extension. Suppose that $L$ is some field which contains both $F$ and $E$ as subfields. I want to show ...
4
votes
1answer
164 views

Showing a polynomial not reducible.

How do I show that $f(x)=1+2x+\cdots+(p-1){x}^{p-2}$ is not reducible on $\mathbb{Q}$, where $p$ is prime.
2
votes
1answer
329 views

All irreducible factors of $p(x)$ in $L[x]$ have the same degree using möbius function

Let $L:K$ be a field extension, and let $p(x) ∈K[x]$ be irreducible and has no zeros in $L$ show that all irreducible factors of $p(x)$ in $L[x]$ have the same degree. Assume that $L:K$ is normal and ...
2
votes
1answer
68 views

If $F\subseteq R \subseteq E$ where $E$ is an extension of $F$ and $R$ is an $F$-subspace, show that $R$ is a field.

I have another abstract algebra question. I stated it in the title, but here it is in more detail: Let $F\subseteq R \subseteq E$ where $E$ is an algebraic extension of the field $F$. If $R$ is an ...
0
votes
1answer
428 views

cube root of 2 not in Q(primitive root)

I am asked to prove that "$\sqrt[3]{2}\not\in\mathbb{Q}(\alpha_n)$ for all $n$ where $\alpha_n$ is the primitive nth root of unity" I have attempted to use contradiction with the tower theorem. I got ...
7
votes
1answer
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 ...
4
votes
2answers
212 views

find all the $n$ such that $ \phi(n) , \phi(n+1) , \phi(n+2)$ are powers of 2

Find all the natural numbers such that, the regular $n , n+1 , n+2 $ gons are constructible. Well this problem can be restated in the following way. Since the construction of the regular n-gon is ...
0
votes
1answer
96 views

a quadratic closure of a field that is closed under conjugation

Definition : Let $K$ be a field with $char(K)=0$ , let's define $ K^{quad}\subset \overline{K}$ by: $$ K^{quad} = \bigcup\limits_{i \geqslant 1} {K_i } $$ where $$ K_1 = K $$ $$ K_{i + 1} = ...
0
votes
3answers
57 views

$x^3-\alpha \in \Bbb Q(\alpha)[x]$ is irreducible

Given $\alpha\in \Bbb C$ trascendental , and such that $|\alpha|=1$ (I don't know if this is necesary but I need only this case). Then I have to prove that the polynomial $x^3-\alpha \in \Bbb ...
0
votes
1answer
50 views

proving that this family of angles, cannot be trisected

Given a field $ \Bbb Q \subset K \subset \Bbb C$. One can prove that $\beta \in \Bbb C$ is constructible over $K$ iff the galois group of the minimal polynomial over $K$, $m_{\beta}(x)\in K[x]$ is a ...
4
votes
3answers
95 views

Rational functions can't take same value infinite times?

Let $K$ be a field of characteristic zero. Can $f/g\in K(X)$ considered as a function $K\to K$ take the same fixed value $\alpha \in K$ infinite times? By elementary calculus this is not the case if ...
3
votes
2answers
285 views

Tensor product of fields

Suppose $D$ is a finite dimensional skew field over the field $K$. Futher, take $x \in D\setminus K$ and let $L=K(x)$. My question: is $D\otimes_K L$ a field? I think not. However I can't seem to ...
1
vote
1answer
105 views

Using Smith Normal Form to prove that even for a singular square matrix with entries in field $F$, $\exists U$ nonsingular where $(UA)^2 = UA$

How might I use Smith Normal Form of a matrix to show that even for a singular square matrix with entries in field $F$, $\exists U$ nonsingular (with entries in $F$) where $(UA)^2 = UA$? The ...
4
votes
1answer
1k views

the discriminant of the cyclotomic $\Phi_p(x)$

I'm very bad in computations of this kind :/. I don't know tricks for computing the discriminant of a polynomial, only the definition and using the resultant, but it's very complicated to do only with ...
1
vote
1answer
85 views

counting the real zeros of a polynomial and proving that it's irreducible over $\Bbb Q$

Let's consider the polynomial $$ f\left( x \right) = \left( {x^2 + 2} \right)\prod\limits_{i = - k}^k {\left( {x - 2i} \right) + 2 \in {\Bbb Q}\left[ x \right]} $$ . Let's suppose that $ p = 2k + ...
1
vote
1answer
37 views

Dependence of $|K|$ and the union of the subspaces.

If $V$ is a $K$-vector space with $|K|\ge n$ and $V_1, \ldots, V_n$ are subspaces of $V$ such that $V=V_1 \cup \cdots \cup V_n$, then $V=V_i$ for some $i$. Is there a counterexample that for $|K|< ...
2
votes
1answer
178 views

Motivation on traces, norms and discriminants

I am looking for some motivation for the definitions of trace, norm and discriminant (in the context of finite field extensions). For example (but not limited to) any interesting theorems proved using ...
7
votes
1answer
132 views

“Real part” of a number field

Let ${\mathbb K} \subseteq {\mathbb C}$ be a finite extension of $\mathbb Q$, and let $n=[{\mathbb K} : {\mathbb Q}]$. Let $X_{\mathbb K}$ denote the set of all “components” (i.e., real and imaginary ...
0
votes
1answer
76 views

Remark in Lang on Extension Fields

This is from Serge Lang's Algebra, in Chapter 5. If $K$ is a field, and $K\subset E$ is a field extension, then for $\alpha\in E$, then $K(\alpha)$ is defined to be the smallest subfield of $E$ which ...
1
vote
2answers
498 views

simple and irreducible radical extension

Definition : Let $K$ be a field. If $\alpha$ is an algebraic element over K, such that $\alpha^n \in K $ and such that $x^n-\alpha^n \in K[x] $ is irreducible over K. Then we call $ K(\alpha)$ a ...
2
votes
1answer
64 views

Construction of a field

Given the polynomial $$f(x)= x^4-16x^2+4$$ which has $a=\sqrt 3+\sqrt 5$ as one of its roots in $\Bbb C$, can you use $f(x)$ to construct a field $E$ of the form $Q[x]/I $ for some appropriate ideal ...
6
votes
1answer
386 views

Degree of field extension

If $p$ is odd prime and $c=\cos(\frac{2\pi}{p})$, $s=\sin(\frac{2\pi}{p})$ then for which values of $p$ does $\mathbb{Q}(s,c)=\mathbb{Q}(c)$?
4
votes
3answers
498 views

The smallest field containing $\sqrt[3]{2}$

Someone could explain how to build the smallest field containing to $\sqrt[3]{2}$.
1
vote
0answers
91 views

Generating de Bruijn sequences over GF(4)

I'm trying to generate a de Bruijn sequence in GF(4) of order $k$ using the recursive formula from this paper: $s_i = \theta_{k-1}s_{i-1} + \theta_{k-2}s_{i-2} + \dots + \theta_{0}s_{i-1k}$ where ...
0
votes
1answer
389 views

$|\mathbb{Q}(e^{2\pi i/n}) : \mathbb{Q}(\cos(2\pi/n)| = 1 \text{ or } 2$

I was trying to follow a sentence in a paper, which says that: $|\mathbb{Q}(e^{2\pi i/n}) : \mathbb{Q}(\cos(2\pi/n)| = 1 \text{ or } 2$ because $e^{2\pi i/n}$ is a root of $x^2 - 2\cos(2\pi/n) + ...
2
votes
1answer
70 views

Powers of $\Bbb Z_p\cap \Bbb Q$ on $1+x\Bbb F_q[[x]]$

Set $q=p^r$ for the finite field $\Bbb F_q$. In the formal power series ring $\Bbb F_q[[x]]$, there is a notion of convergence given by the underlying $(x)$-adic topology. If ...
1
vote
0answers
64 views

$Μ : K$ need not be radical

Show that $Μ : K$ need not be radical, Where $L : K$ is a radical extension in $ℂ$ and $Μ$ is an intermediate field.
2
votes
1answer
456 views

$f$ is solvable by radicals, but the splitting field $L:Q$ not radical extension.

Give an example of a polynomial $f$ over $Q$ which is is solvable by radicals, but the splitting field $L$ for $f$ is not radical extension over $Q$.