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 topic....

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60 views

Proving the roots of a polynomial are a subfield

Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$). What ...
0
votes
1answer
50 views

Find the fixed field of $f\colon K(X)\to K(X)$ given by $f(X)=1/X$

Let $K(X)$ denote the field of fractions of the polynomial ring $K[X]$ over a field $K$. Find the fixed field of the automorphism $f\colon K(X)\to K(X)$ given by $f(X)=1/X$.
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1answer
109 views

Why principal ideal should be commutative?

According to the definition of Principal Ideal it should be commutative. What if the ring is not commutative? Which means $ar\neq ra$ where $a\in I, r \in R$. Does it lead to a contradiction? Because ...
3
votes
1answer
198 views

If $f$ is an irreducible rational polynomial then all the roots over $\mathbb{C}$ are distinct

I'm trying to show that if $f \in \mathbb{Q}[t]$ is irreducible then all the roots of $f$ in $\mathbb{C}$ are distinct. My first issue, am I right in thinking that the roots are distinct iff $hcf(f,f'...
3
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0answers
112 views

Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
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1answer
90 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
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1answer
54 views

Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a $Gal(k_S|k)$-...
5
votes
1answer
470 views

Field extensions of finite degree and primitive elements

Over a field $F$ of characteristic $0$, if every every element of an extension field $K$ has degree less than $n$ over $F$, does this tell us that $K$ is a finite degree extension of $F$? So it would ...
0
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2answers
337 views

Number of intermediate fields in non-separable extensions that are also not purely inseparable

If a field extension is separable then we know there are only finitely many intermediate fields. If a field extension is purely inseparable then it is possible for there to be infinitely many ...
0
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1answer
35 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 ...
1
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3answers
121 views

Automorphisms on a field $F$

I am trying to understand this proposition with respect to algebraic closures of a field $F$ Prop: If $F$ is a finite field, then every isomorphism mapping $F$ onto a subfield of an algebraic closure ...
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votes
1answer
40 views

How we can prove the irreducibility of polynomials [closed]

Suppose $A,B$ are algebraic over $F$ with minimal polynomial $f$ and $g$ respectively. Prove that $f$ is irreducible over $F(B)$ iff $g$ is irreducible over $F(A)$.
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0answers
231 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
2
votes
1answer
77 views

Roots of polynomials in field of prime characteristic p

Let $F$ be a field of characteristic $p$, and let $\alpha \in F$. Let $f \in \mathbb{Z}_p[x]$ be such that $f(\alpha) = 0$. Apparently, we have $f(\alpha^p) = 0$. This was mentioned but not proved in ...
3
votes
3answers
2k views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be $1,(\...
2
votes
3answers
723 views

Example of finite extension which is not finitely generated extension

I just read the theorem Finitely Generated Algebraic Extension is Finite. So a field being finitely generated and algebraic is a sufficient condition for it being finite. Is it also a necessary ...
2
votes
2answers
172 views

Difference between i and -i

Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a - bi$ does the image ...
1
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1answer
211 views

embedding of a finite algebraic extension

In one of my courses we are proving something (so far, not surprising) and using the fact: if $F$ is a finite algebraic field extension of $K$, there is an embedding of $F$ into $K$. Well, doesn't ...
3
votes
0answers
63 views

$\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism $\...
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2answers
171 views

What does the idea of splitting mean when used with fields and polynomials?

i want to understand what does field splitting represent,from my book A Course In Galois Theory by D.J.H Garling this term is explained by following sentences Suppose that $K$ is field, that $f\...
2
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1answer
225 views

Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $ g := x^n - 1 \in F[x]$ Is it true that $ g$ has distinct roots in $F$ if and only if $...
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2answers
168 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 $\cos\theta=\...
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2answers
134 views

$x^5-x^2+1$ is separable over all fields

Prove that $p(x)=x^5-x^2+1$ is separable over all fields. When the field is finite or of characteristic zero it is automatically true, since any polynomial is separable. The definition of ...
4
votes
0answers
69 views

galois extension of imaginary quadratic field

Let $K$ be an imaginary quadratic field, and let $K \subset L$ be a Galois extension. Let $\tau$ denote complex conjugation. Show that $L$ is Galois over $\mathbb{Q}$ if and only if $\tau(L)=L$. My ...
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1answer
93 views

Proposition about intermediate field extensions

This is a problem from Algebra, Hungerford. Exercise V.5.21. (a) Let $L$ and $M$ be intermediate fields of the extension $K \subset F$, of finite dimension over $K$. Assume that $[LM : K:] = [L : K][...
10
votes
1answer
159 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 (...
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1answer
1k views

Prove that the Gaussian rationals is the field of fractions of the Gaussian integers

I'm looking to prove that $\Bbb Q[i] = \{ p + qi : p, q \in \Bbb Q \}$ is the field of fractions of $\Bbb Z[i] = \{p + qi : p, q \in Z\}$. I am familiar with definition of a field of fractions. For ...
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1answer
63 views

Do we have $K(\beta)=K(\beta^2)$ for field extension of odd degree?

Let $K\subseteq K(\beta)$ be a finite field extension of odd degree. Does this imply $K(\beta)=K(\beta^2)$?
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2answers
56 views

Is there an easy way to find irreducibele polynomials in $\mathbb{Q}[x]$ with root exp(2$\pi$ i/k)

Is there an easy way to find irreducible polynomials in $\mathbb{Q}[x]$ with roots $\exp\bigl(\frac{2\pi i}{k}\bigr)$ for $k=0,1,2,\dots,10$?
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2answers
183 views

is an automorphism on the prime field the identity map?

Let $F$ be a field and f an automorphism on K. Is $f$ the the identity map on the prime field of $F$? I feel it should follow from the fact that the prime field is either $\mathbb{Q}$ or $F_p$, but I ...
2
votes
1answer
886 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 ...
1
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0answers
87 views

Factorize polynomial in $\mathbb Z_2[X]$

What is the most efficient way (less time consuming, algorithmically) to factorize polynomials in $\mathbb Z_2[X]$ ? For small degree polynomials, I just try every possibilities (like $X^2+1=(X+1)^2$)...
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1answer
65 views

Transcendental number over $\{k\in K\mid f(k)=k\}$

Let $K$ be a field and $f:K\rightarrow K$ be a ring endomorphism. Prove that if $\alpha\in K\setminus f(K)$, then $\alpha$ is transcendental over the subfield of $K$, $F:=\{k\in K\mid f(k)=k\}$. My ...
2
votes
1answer
54 views

number field:How can i prove that ${\Bbb Q}[\sqrt{-3}]$ is a cyclotomic field?

Can you help me with this ''simple'' exercise: Prove that ${\Bbb Q}[\sqrt{-3}]$ is a cyclotomic field.
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1answer
134 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
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1answer
53 views

Group Automorphism on the smallest field containing roots

Let $L$ be the smallest subfield of $\mathbb{C}$ that contains the solutions to $x^4-2=0$. So $L$ will contain all the rationals. I have in my notes $Aut(L)$ is finite, however, I cannot think of ...
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1answer
46 views

finite field or order $q$

Let $q=p^n$, $p$ is a prime number Let $\Bbb F_q$ a finite field Define function $f$ by $$f:\begin{array}{l} \Bbb F_q \to \Bbb F_q\\ x \mapsto x^m \end{array}$$ $m$ is a natural number Then what ...
4
votes
1answer
198 views

Irreducibility of $p(x)=x^4-4x^2+8x+2$ over $\mathbb{Q}(\sqrt{-2})$- Dummit Foote Abstract algebra $9.4.10$

Question is : Prove that the polynomial $p(x)=x^4-4x^2+8x+2$ is irreducible over the quadratic field $F=\mathbb{Q}(\sqrt{-2})$. [Hint : first use the method of proposition $11$ for the U.F.D $\...
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votes
3answers
62 views

Checking if something is a unit

Check if $\mathbb{Z}_5/x^2 + 3x + 1$ is a field. Is $(x+2)$ a unit, if so calculate its inverse. I would say that this quotient ring is not a field, because $<x^2 + 3x + 1>$ is not a maximal ...
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1answer
24 views

The basis of extension field contains $1$

Let $E$ be a extension of field $F$. It is known that $E$ may be considered as vector space over $F$. Is it always possible to find a basis in $E$ that contains $1$? Maybe it is possible when $E$ is ...
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1answer
70 views

If $K$ is not perfect then there are inseparable irreducible polynomials (Dummit & Foote, P549 )

If $K$ is not perfect then there are inseparable irreducible polynomials. This is not obvious to me at all. I have tried to reduce the question to If $K$ is not perfect then there exist ...
0
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1answer
45 views

If $E$ and $K$ are splitting fields, then $E\cap K$ is a splitting field for what polynomial?

If $E$ and $K$ are splitting fields over a field $F$ for some polynomials $f$ and $g$, then the composite $EK$ is the splitting field for $fg$. It is also standard that $E\cap K$ is a splitting field ...
4
votes
2answers
486 views

Field of order 8, $a^2+ab+b^2=0$ implies $a=0$ and $b=0$.

I was able to come up with a proof for this problem however, it seems like my argument can work for any field of even order and not just odd powers of 2 so I'm convinced there is something wrong here. ...
1
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1answer
220 views

Field of all algebraic reals over $\mathbb{Q}$ has infinite degree [closed]

I am trying to show that field of all algebraic reals over $\mathbb{Q}$ has infinite degree. I guess that $$1,\sqrt{2},\sqrt[3]{2}, \sqrt[4]{2}, ...$$ are lineary independent but can't prove it.
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1answer
39 views

A question about the action of $S_n$ on $K[x_1,…,x_n]$

Let ${K}$be the field ($\,Char\,K\not=0)$. Let $n\in \mathbb{Z}^{+}$. $S_n$ acts on $K[x_1,...,x_n]$in the following way: If $p\in K[x_1,...,x_n]$ and $\sigma\in S_n$, then $\sigma p$ is the ...
4
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3answers
148 views

give an example of algebraic numbers $\alpha, \beta$ such that…

Question is to find algebraic numbers $\alpha, \beta$ such that : $$[\mathbb{Q}(\alpha):\mathbb{Q}]>[\mathbb{Q}(\beta):\mathbb{Q}]>[\mathbb{Q}(\alpha\beta):\mathbb{Q}]$$ It is not so difficult ...
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3answers
221 views

The kernel of homomorphism of a local ring into a field is its maximal ideal?

I have a question about the proof of Theorem 3.2. of Algebra by Serge Lang. In the theorem $A$ is a subring of a field $K$ and $\phi:A \rightarrow L$ is a homomorphism of $A$ into an algebraically ...
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0answers
110 views

Separability of field extensions

I'm trying to figure out if these statements are equivalent for an arbitrary field extension $L/k$, such that $\text{char} (k) = p$ and $A$ is a reduced commutative $k$-algebra. $1)$ $L/k$ is ...
3
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2answers
161 views

The ring of integers of $\mathbf{Q}[i]$

Is there a relatively "simple" (in the sense that it does not require knowledge of algebraic number theory) proof that the ring of integers of the algebraic number field $\mathbf{Q}[i]$ is $\mathbf{Z}[...
1
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1answer
97 views

About the notation $\mathbb{Z}[x]/(f(x),p)$

Let $f(x)\in \mathbb{Z}[x]$ be a polynomial and $p$ be a prime. What does the notation $\mathbb{Z}[x]/(f(x),p)$ mean? Is it $\mathbb{Z}/p\mathbb{Z}[x]/(f(x))$ ?