3
votes
1answer
15 views

different factors of irreducible polynomials over a Galois extension does not share roots

Let $F$ be a field and $E/F$ a galois extension. Let $f\in F[x]$ be an irreducible polynomial. I'm trying to prove that all the irreducible factors of $f(x)$ over $E[x]$ have the same degree. If ...
0
votes
0answers
58 views

My answer to this simple question is wrong but I don't know why

I'm self-studying abstract algebra, and prior to fields there's a brief section on vector spaces. One of the questions asks: "Is $U = \{(a, b-1, c)| a, b, c \in F \}$ a subspace of $F^3$? ($F$ a ...
2
votes
1answer
58 views

Showing $\operatorname{Aut}(\mathbb{R})$ is abelian

I have an exercise (not for a class) that asks whether $\operatorname{Aut}(\mathbb{R})$ (field automorphisms) is abelian. I know that $\operatorname{Aut}(\mathbb{R})$ is just the trivial group, but ...
1
vote
1answer
27 views

Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
0
votes
0answers
43 views

Quotient Field of an Integral Domain

The question is: Let $n$ be a positive integer and let $D$ be the subset of all rational numbers of the form $$\frac{a}{n^k}$$ with $a \in \mathbb{Z}$ and $k$ any positive integer. Show that $D$ ...
4
votes
2answers
40 views

Intermediate fields of Gal($x^5-2$, $\mathbb{Q}$)

I'm having trouble findind the fixed field of each subgroup of the Galois group of $\mathbb{Q(\alpha, \omega)}$, where $\alpha = 2^{\frac{1}{5}}$ and $\omega$ is a 5-th primitive root of unity. I list ...
6
votes
0answers
118 views
+200

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Car}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
0
votes
3answers
84 views

Can $\mathbb{Z}$ be endowed with operations that give it the structure of a field?

Does there exist some definition of addition and multiplication for which the set of all integers is a field?
2
votes
0answers
24 views

Determining if an extension is a normal extension [MORANDI'S BOOK]

I'm dealing with the following problem: Let $K$ be a field and suppose that $\sigma\in\mbox{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K/F$ is algebraic, show that ...
10
votes
1answer
467 views

Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted. For example, I'm studying some Algebra right now. I have found three universal ...
2
votes
1answer
28 views

Using Kronecker's theorem to construct a field with four elements

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...
1
vote
2answers
41 views

Prove the fractional field of an integral domain is the smallest field containing the integral domain

I have two questions about the fractional field of an integral domain. Given an integral domain $D$: Is there a difference between saying "the fractional field of $D$ is the smallest field ...
1
vote
0answers
35 views

Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
1
vote
2answers
27 views

Suppose that $L(\alpha):L:K$ and that $[K(\alpha):K]$ and $[L:K]$ are relatively prime.

Show that the minimal polynomial of $\alpha$ over $L$ has its coefficients in $K$. I tried an approach but I got stuck: We have that the following field extensions: $L(\alpha)/L$ and $L/K$ and we ...
1
vote
1answer
43 views

Extension of an Isomorphism

Suppose $E_1, E_2 \subset E$ are proper subfields. In general, if one has an isomorphism $\sigma:E_1\to E_2$, is it possible to extend it to an isomorphism $\psi:E\to E$ s.t. $\psi|_{E_1} = \sigma$ ...
0
votes
1answer
34 views

Irreducible polynomial iff the condition is satisfied

I am asked to show that the polynomial $f(x)=x^n+1 \in \mathbb{Q}[x]$ is irreducible if and only if $n=2^k$ for any integer $k\geq 0$. Could you give me some hints what I could do to show this??
4
votes
1answer
33 views

Fields extensions over isomorphic fields of different degrees

What are the simplest examples of situations where in a field $F$ there are two subfields $L_1$ and $L_2$ such that extensions $F/L_1$ and $F/L_2$ are finite, degrees are different $$ [F:L_1] \neq ...
0
votes
1answer
46 views

Show that it is a field

$K \leq E$ an algebraic extension. I am asked to show that each subring of $E$ that contains $K$ is a field. I have done the following: $K \leq E$ algebraic $\Rightarrow \forall a \in E, \exists ...
8
votes
2answers
102 views

${\rm Hom}_R(M, R/M) =\{0\} \implies R$ is a field.

Let $R$ be a local ring with maximal ideal $M$. Suppose $M$ is finitely generated. Prove that if ${\rm Hom}_R(M, R/M) =\{0\}$, then $R$ is a field. ${\rm Hom}_R(M, R/M)$ stand for the group of ...
4
votes
1answer
74 views

On a Proof that the Splitting Field of a Separable Polynomial is Galois

Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
0
votes
2answers
30 views

Field extension-degree

I have the following question... $K\leq E$ a field extension. When we have that $$[E:K]=1$$ do we conclude that $K=E$?? Or must also something else be satisfied so that $K=E$ ??
0
votes
2answers
46 views

Field extension-Why does this hold?

$K\leq E$ a field extension, $a\in E$ is algebraic over $K$. Could you explain me why the following holds?? $$K\leq K(a^2)\leq K(a)$$
3
votes
1answer
32 views

When is $0$ mentioned at anytime when talking about Fields, does this mean we are talking about the number $0$, or is it the additive identity?

When talking about fields, such as the field axioms and the theorems that follow, when $0$ is mentioned at anytime, does this mean we are talking about the number $0$, or is it the additive identity?
0
votes
1answer
49 views

How to check field axioms given addition and multiplication tables

I need help with this question, i want to know the exact method of doing it with explanation. i am not able to get around with the logic of it.
2
votes
2answers
36 views

Write it as an element of this ring?

Since the degree of the irreducible polynomial $x^3+2x+2$ over $\mathbb{Q}[x]$ is odd, it has a real solution , let $a$. I am asked to express $\displaystyle{\frac{1}{1-a}}$ as an element of ...
1
vote
1answer
31 views

Galois group of reducible polynomial

I want to find Gaolois group of $(x^3-x+1)(x^2+1)$ over $ \mathbb Q$. The polynomial of degree third is irreducible and has discriminant $-23$ so it's Galois group is $S_3$. Galois group of the other ...
0
votes
1answer
44 views

Algebraic element

$K \leq L, a \in L$ I am looking at the proof that if $a$ is algebraic over $K$, then $K(a)=K[a]$ : We show that $K[a]$ is a field, then we have that $K \subseteq K[a] \subseteq K(a) \subseteq L$. ...
0
votes
1answer
44 views

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$.

Proof/Counterexample: If $z$ is a complex number and $z\notin \mathbb Q$, then $\mathbb Q(z)=\mathbb Q(z^3,z^5)$. First, $\mathbb Q(z)\subseteq \mathbb Q(z^3,z^5)$ would be trivial, right? Then we ...
2
votes
2answers
25 views

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$.

Proving if $a$ and $b$ are positive rational numbers and $\mathbb Q(\sqrt{a})=\mathbb Q(\sqrt{b})$ then $b=ac^2$ for some $c\in \mathbb Q$. I understand that $\mathbb Q(\sqrt{a})$ is the smallest ...
4
votes
2answers
71 views

For what natural numbers $n$ is $\mathbf Z/n\mathbf Z$ $[x]/(x^3+x+1)$ a field?

I recently saw this question in the exam of a first abstract algebra course in my college. It shouldn't be too difficult, yet I can't seem to get the solution. Any ideas on how to tackle this?
0
votes
2answers
53 views

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$.

Proving $(ab)^{-1}=a^{-1}b^{-1}$ where $F$ is a field and $a,b\in F$. One thing to note is $a^{-1}\ne \large\frac{1}{a}$ (same goes for $b$) in this instance as there could be fields where this isn't ...
24
votes
1answer
518 views

Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true: Let $x,y$ be real, irrational ...
3
votes
2answers
69 views

In the theorem is it necessary for ring $R$ to be commutative?

According to the statement of theorem that a commutative ring $R$ with prime characteristic $p$ satisfies $$\begin{align} (a+b)^{p^n} = a^{p^n} + b^{p^n} \end{align}$$ $$\begin{align} (a-b)^{p^n} = ...
1
vote
3answers
22 views

Isomorphism of field extension

Let $F$ be a field. I need to prove that if $\sigma$ is an isomorphism of $F(\alpha_1,...,\alpha_n)$ with itself such that $\sigma|_F = id_F$ and $\sigma(\alpha_i)=\alpha_i$ for $i=1,...n$, then ...
0
votes
1answer
41 views

Question about $\gcd$

Theorem: Let $K$ be an infinite field and let $L:=K(\alpha, \beta)/K$ be a field extension with $\alpha$ algebraic over $K$ and $\beta$ separable over $K$. Then $L = K(z)$ for a certain $z \in L$. ...
1
vote
1answer
39 views

What is the difference between the algebraic function fields and the fields itself

I'm studying this book and I don't understand exactly what's the difference between the algebraic function field $F/K$ and $F$ itself. Thanks
0
votes
1answer
33 views

How to Prove: If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.

I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go ...
0
votes
1answer
30 views

On an example of an inseparable element in a field extension

Put $F := \mathbb{F}_p(t) = \left\{\frac{f(t)}{g(t)} : f(t), g(t) \in \mathbb{F}_p[t], g(t) \neq 0\right\}$. Now in one book the author considers the so-called Frobenius homomorphism $\mathcal{F}:F ...
0
votes
4answers
43 views

Easy question about tower of fields

If $E/F/G$ is a tower of fields and $[E:G]<\infty$, then does $[F:G]<\infty$? I suspect the answer to be "yes", but somehow the fact that a basis of $E$ over $G$ might have vectors in ...
1
vote
0answers
32 views

Denseness of algebraic and transcendental elements in R [NBHM 2014]

Which of the following statements are true? a. Algebraic numbers over Z are dense in R. b. Transcendental numbers over Z are dense in R. Let me write what I did. For a), the concerned set is a subset ...
0
votes
2answers
35 views

Intersection of any family of subfields is itself a subfield

Prove that the intersection of any family of subfields is itself a subfield. In the countable case: Suppose that $\mathscr K$ is a field and consider $(\mathcal K_n)_{n\in\mathbb N}\subset \mathscr ...
1
vote
3answers
171 views

Can we say “commutative ring = field”?

We know the difference between ring (R) and field (F) is that R does not guarantee multiplication is commutative. Now, if considering commutative R, which means (R,.) is a group, can we say: ...
0
votes
0answers
26 views

Field isomorphism and order of elements

I know that group isomorphism preserves order of element but can someone plese tell me does field isomorphism preserves order of elements?
0
votes
1answer
31 views

Prove $\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$

I want to know why the following two are equivalent: $$\mathbb{Q}(\sqrt2,\sqrt2i)=\mathbb{Q}(\sqrt2+\sqrt2i)$$, where $\mathbb{Q}$ is the rational number field, and ...
0
votes
1answer
35 views

When people say, “K is an extension of k with dimension n”, do they mean as an algebra or as a vectorspace?

For instance, consider k(x), (the fraction field of k[x]). k(x) has dimension 2 as an algebra over k, but dimension \omega as a vectorspace over k. Which one are they talking about, and how can I ...
0
votes
2answers
62 views

Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a\cdot a^p$ Let $a\cdot a^p = b\cdot ...
-3
votes
2answers
101 views

What are the root of $x^3 - 2$ $\in \mathbb{R}[x]$? [closed]

From the given polynomial it is evident that we have to find a +ve number in $\mathbb{R}$ such that the cube is 2 if it exists. There is one and that is $\sqrt[3]{2}$. How to find the other roots in ...
1
vote
1answer
39 views

Write Galois group as semidirect-product

Consider the polynomial $ x^7-13 \in \Bbb{Q}(x)$. Find Galois group and write it as a semi-direct product. Edit: So here is what I have done. I found the dimension of the splitting field over ...
3
votes
2answers
88 views

Field that contains $(\mathbb{Z}/p^n\mathbb{Z})^*$

Let $p$ be an odd prime. There exists a field $F$ that contains an isomorphic copy of $(\mathbb{Z}/p^n\mathbb{Z})^*$ as a multiplicative subgroup? Clearly if $n=1$ we can take $F=\mathbb{F}_p$, but ...
2
votes
0answers
24 views

Finding degree of an extension

Find the degree of the field extension $\mathbb{Q}[\sqrt[3]{2},\sqrt[3]{3}]$ over $\mathbb{Q}$. My approach: Call the desired degree $n$. Clearly, $3|n$ and $n\leq 9$. So possible values of $n$ are ...