1
vote
1answer
32 views

How to show the cyclotomic polynomial is irreducible over $\mathbb{R}(T,\sqrt[n]{T})$

I'm trying to solve the following problem. Let $T$ be a transcendental over $\mathbb{R}$. Put $K=\mathbb{R}(T), n\geq3$. Let $L$ be the least splitting field of $X^n-T$. Then, calculate $[L:K]$. I ...
1
vote
1answer
40 views

How to show $\sqrt[3]{X-i}\notin \mathbb{C}(X,\sqrt[3]{X+i})$

I'm trying to show $\sqrt[3]{X-i}\notin \mathbb{C}(X,\sqrt[3]{X+i})$. But this is harder than I expected. Is there any easy way to show this?
3
votes
1answer
68 views

Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem. But it's too difficult for me. Let $\mathbb{C}(T)$ be a function field, and put $L:=\mathbb{C}(T,\sqrt{T^2 + T +1})$, $K:=\mathbb{C}(T^3)$. Let $M$ be a ...
1
vote
1answer
49 views

How to show this is the minimal polynomial

I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism $\sigma, \tau$ of ...
0
votes
2answers
31 views

Is $\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}[X]/(X^2-1)$

Is $$\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}[X]/(X^2-1)$$ I know that $\mathbb{Q}[X]/(X^2+1) \cong \mathbb{Q}(i)$ but I can't say that $\mathbb{Q}[X]/(X^2-1) \cong \mathbb{Q}(1) = \mathbb{Q}$ ...
1
vote
3answers
75 views

$K^\times$ isomorphic to $\mathbb{Z}/2n\mathbb{Z}$ when $K^\times$ is cyclic

Let $K$ be a field so that $K^\times$ is cyclic. Assume $\operatorname{char} K \neq 2$. Prove that $K$ is finite and $K^\times$ is isomorphic to $\mathbb{Z}/2n\mathbb{Z}$ for some $n$. To prove that ...
2
votes
2answers
45 views

$x\in K(t)$ is algebraic over $K$ if and only if $x\in K$ (with $t$ transcendental over $K$)

I already proved the following: Lemma A: Let $K$ be a field. Let $t$ be transcendental over $K$. Let $f \in K[X] \backslash K$. Then $f(t)$ is transcendental over $K$. Proof: Because $t$ is ...
1
vote
2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
1
vote
2answers
48 views

Field extensions and gcd

Let $L|K$ be a field extension and let $u, v \in L$ be algebraic elements over $K$ such that $[K(u):K]=n$ and $[K(v):K]=m$. Show that if $\gcd(m, n)=1$ then $Irr(v, k)$ is irreducible on $K(u)$. ...
2
votes
1answer
35 views

$\mathbb Q$ Field extension

Consider the Field $F = \mathbb Q(2^{\frac 1 3})$, Is $\sqrt 2 \in F$? I'm trying to figure out how to determine that and similar questions, can you give me a hint or some guidance on how to do that? ...
3
votes
1answer
24 views

Prove that $K(\alpha)=K(\alpha^6)$ when $[K(\alpha):K]=2011$

Let $L/K$ be a finite extension and let $\alpha \in L$ so that $[K(\alpha):K]=2011$. Prove that $K(\alpha)=K(\alpha^6)$. My idea is as follows: $K \subset K(\alpha^6) \subset K(\alpha)$, therefore ...
0
votes
3answers
47 views

Prove that $\alpha_{1} ^k+ \alpha_{2} ^k +…+ \alpha_{n} ^k = n$ for $k=0$ and $0$ for $k = 1,2,…,n-1$?

For $n\geq 2$ let $\alpha_{1} + \alpha_{2} +.....+ \alpha_{n} $ be all the nth roots of unity over a field and the roots are not necessarily to be distinct. So we have to prove that $\alpha_{1} ^k+ ...
1
vote
1answer
49 views

Is there a specific method to finding a basis for vector spaces over $\mathbb{Q}$ ?

I am stuck on the first one but there are 5 questions on this so I really need help with the process. If anyone can help with any of the following. i) Find a Basis for the field K = ...
0
votes
0answers
21 views

Exercise 5 .39 page 46 Real and Abstract Analysis [Hewitt and Stromberg]

How to show that every non-Archimedian field contains a subfield algebraically and order isomorphic to $\mathbb{Q}(t)$, where $\mathbb{Q}(t)$ is the field of rational functions with coefficient in ...
4
votes
1answer
58 views

Subgroups of $F^*$ are cyclic

Q: If $F$ a field then every finite subgroup of $F^*$ is cyclic. Solution: Suppose $d\mid |G|$ for $G$ subgroup of $F^*$ and $G$ not cyclic. Suppose $A,B$ subgroups of $G$ of order $d$. Then ...
0
votes
1answer
57 views

The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
1
vote
0answers
32 views

Function Field of Degree 3 Ramified at 1 and -1

This question is a homework problem, and I'm having a lot of trouble with it. (a) Determine the number of isomorphism classes of function fields K of degree 3 over $F = \mathbb{C}(t)$ that are ...
4
votes
1answer
31 views

Finding the conjugates in $\mathbb{C}$ of a given number over a given field…

I'm having somewhat of a difficult time understand what's being asked—and thus having a hard time answering the question: Find all conjugates in $\mathbb{C}$ of the given number over the given ...
1
vote
2answers
77 views

Proofs field axioms

I'm terrible with these kinds of proofs and this might be a duplicate but I didn't find anything. I need to prove that $\forall x,y \in \mathbb{R}: -(-x)=x$ and $-(x+y) = -x - y$ I know that this is ...
6
votes
4answers
135 views

If a polynomial $g$ divides $f$ and $f'$, then $g^2$ divides $f$?

Here's a homework problem from Artin's Algebra that I'm having a lot of trouble with Let $f(x) \in F[x]$ (where $F$ is a field of characteristic $0$). If $g$ is an irreducible polynomial that is ...
0
votes
1answer
98 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
1
vote
1answer
108 views

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$?

How to find a primitive element of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$? I think that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}):\mathbb{Q}] = 8$, but not really sure how to ...
4
votes
3answers
145 views

Why must a field with a cyclic group of units be finite?

Let $F$ be a field and $F^* \subseteq F$ be its group of units. If $F^*$ is cyclic show that $F$ is finite. I'm a bit stuck. I know that I can represent $F^* = \langle u \rangle$ for some $u \in F^*$ ...
0
votes
3answers
114 views

showing $Q[\sqrt 2] = Q(\sqrt 2)$

The question came in my exam. $Q[\sqrt 2] = \{ a + b \sqrt2 \;| a,b \in Q\}$ and $Q(\sqrt 2)$ is minimal subfield of it's extension containing $Q$ and $\sqrt 2$. (In my book) It calls $F(a)$ ...
2
votes
2answers
45 views

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5. I will just add that this task is slightly ahead of my knowledge of field theory. So any pointers would be ...
0
votes
3answers
81 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
2
votes
2answers
138 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
3
votes
2answers
168 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. ...
2
votes
1answer
57 views

Galois group of intermediate fields

Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$. I've found that there are 2 intermediate fields ...
1
vote
0answers
38 views

Elementary Field Theory: Extension Field of Degree 2

I'm trying to do/understand the following exercise: "Let $E$ be a finite extension of a field $F$. If $[E:F] = 2$, show that $E$ is a splitting field of $F$."* Background: Just beginning ...
0
votes
1answer
25 views

Splitting Fields and Non-constant Polynomials.

I had a question I was stuck on: Let $p(x)$ be a non-constant polynomial of degree $n$ in $F[x]$. Prove that there exists a splitting field $E$ for $p(x)$ such that $[E : F] \leq n!$. My start: By ...
0
votes
0answers
42 views

Why is this not trivial? - $f(x)~|~g(x) \iff g(x) \in \langle f(x) \rangle$

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
0
votes
1answer
23 views

irreducibility of $x^2-a$ in $\mathbb{Z}_2[a]$

Let $a$ be transcendental over $\mathbb{Z}_2$ and let $F=\mathbb{Z}_2(a)$. Prove $p(x)=x^2-a$ is irreducible over $F$ I've been trying to understand this for a while now, but I'm ...
0
votes
2answers
48 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
1
vote
1answer
74 views

Exercise about field extensions [duplicate]

Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...
2
votes
3answers
65 views

Simple field extension inequality proof

Let $\alpha \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F\subseteq \mathbb{C}$ be a subfield. Prove that $[F(\alpha):F]\leqslant [\mathbb{Q}(\alpha):\mathbb{Q}]$. This looks like a problem ...
1
vote
1answer
45 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
0
votes
1answer
27 views

Ideal of a field

Let $F$ be a field. Show that $S$ be a non empty subset of $F^{n} $ then $ I(S) =$ { $ f(x) \in F[x] \hspace{0.1in} \vert \hspace{0.1in}f(s) = 0 \hspace{0.1in} \forall s \in S $ } is an ...
9
votes
1answer
132 views

Set of elements of degree $2^n$ over a base field is itself a field

Let $F \subset L$ be two fields, and define $K = \{\alpha \in L\mid [F(\alpha): F] \text{ is a power of 2} \}$. Our problem is to prove that $K$ is a field. Closure under reciprocation is easy ...
1
vote
1answer
365 views

Prove that the degree of the splitting field of $x^p-1$ is $p-1$ if $p$ is prime

I came across this question I couldn't figure out how to solve: If $p$ is a prime number, prove that the splitting field over $F$, the field of rational numbers, of the polynomial $x^p-1$ is of ...
0
votes
1answer
38 views

$[L_1 L_2 : k] = [L_1 : k] [L_2 : k]$ for two finite field extensions $L_1/k$ and $L_2/k$ with $L_1 \cap L_2 = k$

I want to prove or disprove the following: Let $L_1$ and $L_2$ be two finite extensions of field $k$ inside of an extension $L/k$. Moreover, $L_1 \cap L_2 = k$. Then the degree of the ...
6
votes
3answers
93 views

Find minimal polynomial of this element?

Let $f(x)=x^3+x+1$. $\alpha_1, \alpha_2, \alpha_3 - $ roots of $f$. The task is to determine the minimal polynomial of $\frac{\alpha_1}{\alpha_2}$ over $\Bbb Q $ and $\Bbb Q(\alpha_1)$. My thoughts ...
4
votes
1answer
67 views

$\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??

Let $p,q$ be primes, $p≠q$, then I have to show that $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ So far I've tried a lot of things with minimal polynomials and bases, ...
0
votes
0answers
25 views

Prove that if $f$ is a primitive polynomial over $F_q$ then $f$ divides $Q_{q^m-1}$.

I am not writing my complete proof, and my conclusion is that since all the roots of $f$ are primitive $(q^m-1)st$ roots of unity and so are the roots of $Q_{q^m-1}$. Therefore, $f$ must divide ...
0
votes
1answer
69 views

Locally finite field

Definition: A field is locally finite if every its finitely generated subfield is finite. Show that a field $K$ is locally finite iff it is embeddable to the algebraic closure of $F_p$, for some ...
0
votes
0answers
76 views

Irreducibility of $x^{p}-x-a$ [duplicate]

Let $K$ be a field, and $a \in K$. Show that if the polynomial $x^{p}-x-a$ has no roots in $K$ then it is irreducible over $K$. ( $\operatorname{char}(K)=p$ ) I will use this fact to prove that ...
9
votes
1answer
137 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 ...
7
votes
1answer
79 views

Short method to prove irreduicibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$?

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$? In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...
0
votes
1answer
136 views

Conjugate isomorphism of the automorphism groups of two field extensions?

This is the problem I'm trying to solve: Let $K/F$ be a field extension, and let $f: K → K'$ be an isomorphism of $K$ with a field $K'$ which maps $F$ onto the subfield $F'$ of $K'$. Prove that ...
1
vote
1answer
65 views

Addition in finite fields

For a question, I must write an explicit multiplication and addition chart for a finite field of order 8. I understand that I construct the field by taking an irreducible polynomial in $F_2[x]$ of ...