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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4
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0answers
45 views

Is there a recurrence formula for finding primitive polynomials in order to construct $\mathbb{F}_{p^n}$?

(In a previous question, I asked some related question but it was very disorientingly formulated. I apologize for that.) Let $p$ be a prime and $n\geq 1$ an integer. The finite field ...
0
votes
0answers
21 views

Embedding of splitting field for a family of polynomials

STATEMENT: Let $K$ be a splitting field for the family $\left\{f_i\right\}_{i\in I}$ and let $E$ be another splitting field. Any embedding of $E$ into $K^a$ inducing the identity on $k$ gives an ...
0
votes
1answer
28 views

Verification of proof that for a,b in ring R, assuming ab is a zero divisor at least one of a and b is zero divisor

I'm not so sure if this is correct but here's what I have so far: ab is a zero divisor iff there is a c$\neq$0 s.t. (ab)c=c(ab)=0 given ab$\neq$0 and c$\neq$0. Then we have ...
2
votes
0answers
49 views

Degree of the field extension for $x^4+3$

I want to find a splitting field for : $x^4+3 \in \mathbb{Q}[x]$ and the degree of that extension , I would be thankful if you verify/correct what I have tried for this problem : First I find the ...
1
vote
1answer
15 views

separable polynomial

How to show that if $K$ is a field of characteristic $p$ with $p$ prime and if $f(X)\in K[X]$ is an irreducible and inseparable polynomials, therefore there exist a $d\in\mathbb N, d>0$ such ...
2
votes
2answers
38 views

show that additive group of field of characteristic 0 is not cyclic

Show that additive group of field of characteristic 0 is not cyclic. If it is so then the additive group will be isomorphic to $\Bbb Z$ from here how do I proceed. I have seen Why must a field with ...
2
votes
1answer
42 views

More general constructible numbers?

I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is ...
1
vote
3answers
80 views

Calculate the dimension of the field extension $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}]$

I've though that $[\mathbb{Q}[ \sqrt2 , \sqrt3] : \mathbb{Q}] = [\mathbb{Q}[ \sqrt2] : \mathbb{Q}].[ \mathbb{Q}[\sqrt2, \sqrt3]:\mathbb{Q}[ \sqrt2] ] $ And I know how to prove $[\mathbb{Q}[ \sqrt2] : ...
0
votes
0answers
35 views

Example of a local field of positive characteristic?

I am looking for a local field of positive characteristic, like $Q^{2}_{2}$ was used in this article: in fact, i need an another Example of a local field of positive characteristic like $Q^{2}_{2}$ ...
3
votes
1answer
49 views

$x^{p^2-1}-1$ is divisible by $x^8-1$ when $p$ is odd?

In the proof of reducibility of $x^4+1$ over $F_p$ (which is stated as a corollary of the structure theorem of the finite field $F_{p^n}$), the following implication is used in the Algebra by Dummit ...
1
vote
2answers
31 views

Which general methods of field construction do we know?

This question is partially motivated by: http://mathoverflow.net/questions/3003/in-what-sense-are-fields-an-algebraic-theory whereas I am not a specialist in field theory. Q1: Which methods do we ...
3
votes
1answer
73 views

Prove that any subfield of $\mathbb C$ must contain $\mathbb Q$

I just started reading Linear Algebra by Hoffman and Kunze, and I came across the following line: The interested reader should verify that any subfield of $\mathbb C$ must contain every ...
1
vote
1answer
28 views

What exactly is $k\left(T_{n}\right)_{n\in\mathbb{N}}$?

Let $k$ be a field and $T_{n}$ indeterminates over $k$. Is $k\left(T_{n}\right)_{n\in\mathbb{N}}$ the field of fractions of the form $x=\frac{p}{q}$, where $p\in k\left[T_{i}\right]_{i\in\mathbb{N}}$ ...
0
votes
1answer
24 views

Algebraically closed fields minimal

STATEMENT: This is a portion from Lang's proof of theorem 2.8 in chapter V section 2. If $E$ is algebraically closed, and $L$ is algebraic over $\sigma k$, then $\sigma E$ is algebraically closed and ...
0
votes
3answers
77 views

Why is $\alpha$ transcendental over $K$

Let $K$ be a field and let $\alpha$ be an element of the field $K(T)$ of rational functions, with $\alpha\not\in K$. Prove that $\alpha$ is transcendental over $K$. In this case $\alpha$ is of ...
1
vote
0answers
27 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
1
vote
3answers
41 views

Minimal Polynomial of $a +b\sqrt{2}$ as a function of a, b ∈ $\mathbb Q$

Determine the minimal polynomial over $\mathbb Q$ of $a +b\sqrt{2}$ as a function of a, b ∈ $\mathbb Q$. Let $x=a+b\sqrt{2}$ If $b=0$ then the minimal polynomial is $x-a$ if not, then ...
1
vote
1answer
33 views

Finite field extensions and minimal polynomial

I want to show the following statement: Let L/K be a finite field extension with $[L:K]=p$ for a prime $p$ Show that $[L:K]$ is simple Proof: 1) Choose $\alpha\in L$ with $\alpha \notin K$. Then ...
2
votes
1answer
70 views

Which one is a field?

Which one is a field? i) $\cfrac{\mathbb{Z}[x]}{\langle{x^2+2}\rangle}$ ii)$\cfrac{\mathbb{Q}[x]}{\langle{x^2-2}\rangle}$ I think both are correct because for both the cases ...
4
votes
5answers
152 views

Determine the minimal polynomial of $\sqrt 3+\sqrt 5$

I am struggling in finding the minimal polynomial of $\sqrt{3}+\sqrt{5}\in \mathbb C$ over $\mathbb Q$ Any ideas? I tried to consider its square but it did not helped..
0
votes
0answers
38 views

Degree of extension

Let $a, b \in \mathbb{C}$ and let $[\mathbb{Q}(a) : \mathbb{Q}]=m$, $[\mathbb{Q}(b) : \mathbb{Q}]=n$. Show that $[\mathbb{Q}(a,b):\mathbb{Q}]\leq mn$. If $(m,n)=1$ show that ...
0
votes
1answer
32 views

Splitting field of polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_7[x]$ and let $E \subseteq \overline{\mathbb{Z}}_7$ be the splitting field. Let $F \subseteq \overline{\mathbb{Z}}_7$ the splitting field of the ...
2
votes
1answer
42 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
0
votes
1answer
40 views

Show that $p=2^k+1$

When $p$ is an odd prime and $a=Re \left ( e^{\frac{2 \pi i}{p}} \right)$ then $[\mathbb{Q}(a) : \mathbb{Q}]=\frac{p-1}{2}$. Let $\theta = \frac{2 \pi}{p}$. If $\sin{\theta}$ is a constructable ...
1
vote
1answer
22 views

Show that the equation has exactly $m$ different roots in the algebraic closure


Let $n=p^rm$, where $p$ is a prime, $m \in \mathbb{N}, r \geq 0$ an integer and $(p,m)=1$. 
I have to show that the equation $x^n=1$ has exactly $m$ different roots in the algebraic closure ...
1
vote
1answer
37 views

Element from a formal power series that is algebraic over a field

I don't know how I should solve this exercise: The polynomial ring $K[T]$, and hence also its field of fractions $K(T)$, is a subring of $K((T))$. Give an example, for some field $K$ , of an ...
0
votes
1answer
29 views

Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$

Prove that: $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$ We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ...
0
votes
0answers
29 views

Symmetric group and fields

Let $K$ a field and $E=K(X_1,...,X_n)$ the fraction field of the domain of the polynomials $K[X_1,...,X_n]$. 1) Show that the symmetric group $S_n$ is a group of automorphism of $E$ 2) Show that ...
0
votes
1answer
11 views

Separable extensions-Need help

Let $K \leq M \leq E$ be field extensions, with $K \leq E$ separable. Show that the extensions $K \leq M$ and $M \leq E$ are separable. The extension $K\leq E$ is separable if all the elements in ...
-1
votes
2answers
50 views

If $\overline{f}(x)$ is irreducible in $\mathbb{Z}_p[x]$ then $f(x)$ is irreducible in $\mathbb{Z}[x]$

Let $f(x)=a_0+\dots +a_n x^n \in \mathbb{Z}[x]$. Let $p$ be a prime with $p \nmid a_n$. We define $\overline{f}(x)=\overline{a_0}+\dots +\overline{a_n} x^n \in \mathbb{Z}_p[x]$ How can I show that ...
0
votes
1answer
17 views

What is the relation between $Irr(a, F)$ and $Irr(a, K)$?

We have that $F \leq K \leq L$ and $a \in L$. If $a$ is algebraic over $F$ then it is also algebraic over $K$. What is the relation between $Irr(a, F)$ and $Irr(a, K)$? Let $Irr(a, K)=p(x) \in K[x]$ ...
0
votes
2answers
71 views

$\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$

I have to show that $\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$ and then I have to find $Irr(\sqrt{5}, \mathbb{Q})$. How can I show that $\sqrt{5} \in \mathbb{R}$ is algebraic over ...
0
votes
2answers
28 views

Why does it stand that #$\mathbb{Z}_p(a)=p^n$?

If $\mathbb{Z}_p \leq K$ an algebraic extension, then $K$ has the identity $$\forall a \in K, \exists b \in K \text{ with } a=b^p$$ The proof is the following: Let $a \in K$. We take $\mathbb{Z}_p ...
2
votes
1answer
54 views

Real subfield of cyclotomic field is generated by $\zeta+\zeta^{-1}$

Let $p\neq 2$ a prime number, $\zeta=e^{\frac{2i\pi}{p}}$ and $\alpha=2\cos\left(\frac{2\pi}{p}\right)$. We consider the field extension $F=\mathbb Q(\zeta)$ and $E=F\cap \mathbb R$ of $\mathbb Q$. I ...
0
votes
1answer
22 views

How do we find the embeddings?

In my notes there is the following example: $$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id ...
2
votes
1answer
48 views

Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
2
votes
1answer
34 views

Compute degree and Galois group of a Galois extension

I'm trying to show that $[L:\mathbb{Q}]=8$, where $L=\mathbb{Q}(i, \sqrt{2}, \sqrt{3})$. I tried using the tower law to show this by saying: $[L:\mathbb{Q}]=[L:\mathbb{Q(\sqrt{2}, ...
0
votes
1answer
30 views

How to find the splitting field?

How can I find the splitting field of $x^n+1 \in \mathbb{Q}[x]$ ?? If we have for example, $x^n-1 \in \mathbb{Q}[x]$ we would do the following: $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \dots +x+1)$$ So, the ...
1
vote
1answer
38 views

Field at which $f(x)$ splits [closed]

Let $f(x) \in \mathbb{Z}_p[x]$. Show that there is a finite field $\mathbb{F}_{p^n}$ at which $f(x)$ splits. And if $f(x)$ is also separable, show that $f(x) \mid x^{p^n}-x$. Could you give me some ...
0
votes
1answer
51 views

The union of finite field extensions is a finite field extension

Assume that all elements under discussion are algebraic over $F$. Let the notation "$K=F(A)$" mean that $A\subseteq K$ and there is an injective homomorphism $\sigma:F\to K$, and every element of $K$ ...
2
votes
4answers
137 views

Roots of different irreducible polynomials are algebraically independent

Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$. Let $\alpha$ be a root of some other monic irreducible $g\ne f$. Then is $f$ still irreducible in $F(\alpha)$? Is it true ...
1
vote
0answers
21 views

Field extensions and quotient fields

STATEMENT: Suppose that $F\subseteq E$ is a field extension of $F$. And assume $u\in E$ is transcendental over $E$. Then it readily follows that $F(u)\cong F(x)$, where $F(x)$ is the quotient field of ...
1
vote
2answers
34 views

Find the splitting field of a polynomial

The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, ...
0
votes
1answer
33 views

Field of rational functions over $\mathbb{F}_p$

Let $K=\mathbb{Z}_p(x,y)$ be the field of rational functions of variables $x,y$ with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. Let $g(t)=t^p-x, h(t)=t^p-y \in K[t]$ and $E$ is the ...
0
votes
0answers
38 views

Show that the fields are equal

I have to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$. $$$$ To show this do I have to take first an element of ...
1
vote
1answer
30 views

Basis for field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$

I'm trying to find a basis for the field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$, where $\zeta$ is the cube root of unity. I attempted this with starting with a set of elements I know ...
2
votes
3answers
217 views

Degree of field extension is infinite

If we have the field extension $\mathbb{Q}\leq \mathbb{R}$, could you explain me why it stands that $[\mathbb{R}:\mathbb{Q}]=+\infty$ ??
2
votes
1answer
28 views

Multiplicative group of a field contains maximal n-1 elements with order n

Let $F$ be a field and $n\in \mathbb N,n>1$. I want to show that the multiplicative group $K$\ $\{0\}$ contains maximal $n-1$ elements with order $n$. I actually don't have any ideas how to solve ...
0
votes
1answer
49 views

Galois group of a polynomial and subfields

Suppose $f(x)\in \mathbb{Z}[x]$ is an irreducible quartic whose splitting field has Galois group $S_4$ over $\mathbb{Q}$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$. a) Prove ...
1
vote
1answer
31 views

This is only a subspace if $b=0$ - Axler - LADR p13

I have written here in Axler - Linear Algebra Done Right, page $13$. If $b\in \mathbb{F}$, then $\{(x_1,x_2,x_3,x_4)\in \mathbb{F}^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb{F}^4$ if and only ...