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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1answer
78 views

Algebraic extension and infinite polynomial ring

Let $K$ a field and $F=K(\alpha_i : i\in I)$ an algebraic extension of $K$. Is it true that for all $z\in F$ there exists $i_1,\dots,i_k\in I$ such that $z\in K(\alpha_{i_1},\dots,\alpha_{i_n})$ ? ...
1
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1answer
232 views

Ring homomorphism between two field extensions that is the identity over ground field

I have been going through some problems in field theory recently, and problem that I came across was the following: "Give an example of (or show it is not possible to have) a field extension ...
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2answers
157 views

Finite Field Extensions

I'm not terribly familiar with field theory, so I'd appreciate some help understanding this homework problem. For context, this problem is for an algebraic geometry course, in relation to function ...
1
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2answers
250 views

An element not in a field extension [duplicate]

Possible Duplicate: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? Consider the field extension $\mathbb{Q}(\sqrt2)$. I want to show that $\sqrt5 \notin \mathbb{Q}(\sqrt2)$. If this ...
2
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1answer
277 views

Number of injective field homomorphism

Let $F$ and $L$ be two field extensions of $K$. The extension $F/K$ is finite of degree $n$ and the extension $L/K$ may be infinite also. I want to count the number of injective homomorphism from ...
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2answers
217 views

Is it true that if $K/F$ is a splitting field and $\phi \colon K \to K'$ is an isomorphism fixing $F$ then $K = K'$?

Let $K$ and $K'$ be finite extensions over a field $F$ such that $K$ is a splitting field for a polynomial $p(x)$ over $F$, and let $\varphi \colon K \to K'$ be an isomorphism which fixes $F$. ...
2
votes
2answers
278 views

Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.

I'm hoping someone can give me a nudge in the right direction... Let $F$ be a finite field, and let $f(x)$ be a nonconstant polynomial whose derivative is the zero polynomial. Prove that $f$ ...
2
votes
2answers
132 views

Repeated roots for polynomial in $\overline{ \mathbb F_{p}}$

Let the $ \mathbb F_{p}$ denote the finite field of $\mathbb Z/ p \mathbb Z$ and $\overline{ \mathbb F_{p}}$ its algebraic closure. Now let $f(x)=X^p- b \in \overline{ \mathbb F_{p}}[x]$. I want to ...
3
votes
1answer
628 views

Field extension, primitive element theorem

I would like to know if it is true that $\mathbb{Q}(\sqrt{2}-i, \sqrt{3}+i) = \mathbb{Q}(\sqrt{2}-i+2(\sqrt{3}+i))$. I can prove, that $\mathbb{Q}(\sqrt{2}-i, \sqrt{3}+i) = ...
13
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3answers
1k views

Inseparable, irreducible polynomials

The standard examples of irreducible, inseparable polynomials that one encounters in an introductory course on field theory all seem to have only a single root in an algebraic closure. Are there ...
0
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1answer
762 views

The field of rational functions in $n$ variables is a Galois extension of the field of symmetric rational polynomials in $n$ variables.

I've been doing a little bit of field theory for number fields but not much with function fields. The question originally asked says "For some field F, show that the field $F(u_1,\ldots, u_n)$ is a ...
5
votes
1answer
477 views

Realizing $S_n$ as a Galois group

My question is about the realization of the symmetric group $S_n$ as a galois group of a real and normal field extension $K/\mathbb Q$. As I read, such a field $K$ can be obtained as the splitting ...
4
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2answers
794 views

Exercise on separable polynomials over fields of prime characteristic

Having learned about separable polynomials today in class, I tried to do the following exercise concerning separable polynomials, namely: Suppose $f$ is the minimal polynomial of $a$ over a field ...
1
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2answers
735 views

Notation for fields

I'm trying to get caught up wit my abstract algebra class, but I'm getting lost with notation . Can someone help explain these things to me regarding fields? $\mathbb{Q}$ is the field of all rational ...
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3answers
493 views

Subextension of a finitely generated extension of fields

If $E/K$ is a finitely generated field extension and $F$ is an intermediate field how can I prove that $F/K$ is finitely generated?
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2answers
45 views

How to change degree of elements in a field

I'm working through an assignment and need to write several elements as polyomials of degree <= 2. ($x^2+1$)($x+1$) within the field $\mathbb Z$$_3$[x] / ($x^3 + 2x + 1$) And am unsure ...
3
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2answers
283 views

Field extensions described as a matrix

I'm reading Abstract Algebra and there is an exercise I don't understand : The exercise is $Q19.b$ : I just don't understand what the matrix the book talks about ? I can think of something like to ...
2
votes
3answers
370 views

Showing that $\sqrt5$ is not in $\mathbb{Q}(\sqrt7)$

How can I prove that $\sqrt5$ is not in $\mathbb{Q}(\sqrt7)$ ? I can only think of trying to write $\sqrt5 = a+b\sqrt7$ (where $a,b$ are in $\mathbb{Q}$), but I can't think of a good reason that ...
4
votes
4answers
427 views

Finding the minimal polynomial of $\sqrt[3]{7-\sqrt{3}}$ over $\mathbb Q.$

By simple algebraic means I got that $P(x):=(x^3-7)^2-3$ is a polynomial s.t $P(\alpha)=0$ where $\alpha = \sqrt[3]{7-\sqrt{3}}$. I wish to show that $P$ is of minimal degree, is this proof ok ? ...
1
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3answers
183 views

g.c.d of irreducible polynomial and a polynomial of smaller degree

I am reading the book abstract Algebra and the book claims that if $p(x)$ is irreducible polynomial over a field $F$ and $g(x)$ is polynomial of smaller degree then $\gcd(p,g)$ is invertible. for ...
1
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1answer
288 views

Why this extension is purely inseparable?

A field $F$ is separably closed if whenever $\alpha\in\bar{F}$ is separable over $F$ we have $\alpha\in F$. A separable closure of $F$ is a field $E\supset F$ such that $E$ is separably closed and ...
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2answers
338 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
3
votes
1answer
135 views

If $H$ is a group of automorphisms of a field $E,$ is it always true that $(E:E^H)=\operatorname{card}(H)?$

I have recently learned the following theorem. Let $E$ be a field and let $H$ be a finite group of automorphisms of $E.$ Then $(E:E^H)=\operatorname{card}(H),$ where $E^H$ is the subfield of ...
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1answer
1k views

Finding a basis for the field extension $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$

This is exercise A4 in chapter 29 from Pinter's A Book of Abstract Algebra. It is not homework but hints/roadmap would be preferred to a full solution for now. First some context The book works out ...
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5answers
5k views

Finding inverse of polynomial in a field

I'm having trouble with the procedure to find an inverse of a polynomial in a field. For example, take: In $\frac{\mathbb{Z}_3[x]}{m(x)}$, where $m(x) = x^3 + 2x +1$, find the inverse of $x^2 + ...
6
votes
1answer
924 views

Elementary field theory, field extensions of the rationals of degree 2

I've just started some reading and doing exercises on field theory with Galois theory in scope, and have had some trouble with this exercise. I think I have simply misunderstood some of the ...
0
votes
1answer
420 views

Cyclotomic extensions over $\mathbb{Q}$

Let $q(n)$ denote the primitive $n$th roots of unity and let $K=\mathbb{Q}(q(n))$ be the associated cyclotomic field. Let $a$ denote the trace of $q(n)$ from $K$ to $\mathbb{Q}.$ How to prove ...
5
votes
1answer
834 views

Determine the irreducible polynomial for $\alpha=\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}(\sqrt{10})$

I've already found that the irreducible polynomial of $\alpha$ over $\mathbb{Q}$ is $x^4-16x^2+4$. I've also found that $\mathbb{Q}(\sqrt{3}+\sqrt{5})=\mathbb{Q}(\sqrt{3},\sqrt{5})$ and that ...
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2answers
526 views

Derivations in a ring. What applications do they have outside algebra?

INTRODUCTION Let $R$ be any ring, possibly without unity. We call a function $d:R\longrightarrow R$ a derivation on $R$ if it satisfies the following conditions. $(1)$ It is an endomorphism of the ...
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0answers
63 views

Algebraic extensions and polynomial evaluation [duplicate]

Let $F\subset K$ be a field extension and $\alpha_1,\dots,\alpha_n\in F$ algebraic elements over $K$. Is it true that $$K(\alpha_1,\dots,\alpha_n)=K[\alpha_1,\dots,\alpha_n]\ \text{?}$$ Indeed, ...
3
votes
4answers
382 views

Example of fields that are not subsets of the complex numbers

I've read the axioms of a field. To understand the generality of the axioms, could you give me an example of a field which is not (isomorphic to) a subset of complex number (with or without modulus ...
0
votes
1answer
135 views

Find the product of all quadratic irreducible polynomials of $\mathbf{Z}_{3}[x]$?

I'm requested to find the product of all quadratic irreducible polynomials of $\mathbf{Z}_{3}[x]$ . How can I find them ? brute force ? check that every polynomial has no roots ? Or , if I take for ...
2
votes
1answer
431 views

Separable elements in a finite characteristic field

Let $k$ be a field of characteristic $p\ne 0$. Let $K$ be a finite extension of $k$. Let $\alpha \in K$. How do I prove that either ${\alpha}^{p^n}\in k$ for some $n$ or there exists an integer $m$ ...
3
votes
3answers
535 views

Minimal polynomial of $\sqrt[3]{7-\sqrt{2}}$

I want to find the minimal polynomial (over $\mathbb{Q}$) of: $k:=\sqrt[3]{7-\sqrt{2}}$. With simple 'tricks' I got that: $P=(x^3-7)^2+2$ is a polynomial such that $P(k)=0$. But I don't know if, or ...
5
votes
2answers
252 views

Let K/F be a finite extension, given a polynomial in K[x] find another so that their product is in F[x]

Let $K$ be a finite extension of a field $F$, and let $f(x)$ be in $K[x]$. Prove that there is a nonzero polynomial $g(x)$ in $K[x]$ such that $f(x)g(x)$ is in $F[x]$. Should I do this by induction ...
2
votes
1answer
149 views

A question with an odd hypothesis.

Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the ...
5
votes
1answer
112 views

Verifying properties of a discrete valuation.

EDIT: Don't think about this. The problem statement is flawed. (See comments) Let $K$ be a field, and let $K(T)$ be the quotient field of polynomials over $K$. Then I define $v(f/g) = \deg(f) - ...
0
votes
1answer
69 views

“Converse” of dimension counting formula for fields

We recently learned in class that if we have a tower of fields $$F \subseteq E \subseteq L$$ then $[L:F]$ is finite iff both $[E:F]$ and $[L:E]$ are. In this case we have $[L:F]= [L:E][E :F]$. So ...
3
votes
2answers
314 views

Find the first 4 elements of a given field

The polynomial $x^4 + x +1$ is unsplittable under $\mathbb{Z}_2$ . Given the following $K$: $K = \mathbb{Z}_2[x] / \mathbb{Z}_2[x] (x^4 + x +1)= {{a+bx+ cx^2 + dx^3 : a,b,c,d \in \mathbb{Z}_2}} $ ...
1
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1answer
446 views

A problem on field extension

Is there any example of a field extension $K/F$ where degree of extension $[K:F]$ is finite but the number of intermediate field is infinite ?
2
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2answers
135 views

Why is this corollary a corollary? (Field extensions and symmetric polynomials.)

In Stewart and Tall's book on Algebraic Number Theory, they give a theorem of Newton: Theorem 1.12. Let $R$ be a ring. Then every symmetric polynomial in $R[t_1, \ldots, t_n]$ is expressible as a ...
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2answers
154 views

Help proving/starting proof for irreducibility in $\mathbb{Z}_{5}[\sqrt{2}](x)$

I need help proving that $x^2+x+1$ is irreducible in $\mathbb{Z}_{5}[\sqrt{2}](x)$. Anyone be willing to at least help me get a good start? --edit: typo, added the (x) for ...
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3answers
272 views

Proving that $-a=(-1)\cdot a$

As the title reveals, I want to prove (based on the axioms of field) that $$-a=(-1)\cdot a$$ I've been trying for a while now, but couldn't think of a way to do it and it got me thinking that maybe ...
0
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1answer
837 views

Proving an algebraic identity using the axioms of field

I am trying to prove (based on the axioms of field) that $$a^3-b^3=(a-b)(a^2+ab+b^2)$$ So, my first thought was to use the distributive law to show that $$(a-b)(a^2+ab+b^2)=(a-b)\cdot a^2+(a-b)\cdot ...
3
votes
3answers
408 views

Field extension and irreducibility

Let $k$ a field, $P \in k[X]$ irreducible of degree $n \geq 2$, $K$ an extension field of $k$ with degree $m$ such as $\gcd(m,n) =1$. How can I show that $P$ stays irreducible over $K$ ? Thank for ...
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2answers
382 views

Question about the definition of a field…

Just out of curiosity - when we define a field, why bother mention multiplication, when its nothing more then repeating the same addition operation? Here's the definition we were taught in calculus ...
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2answers
367 views

Ring theory, field of fractions

Let $R$=$\mathbb{F}$$[[x]]$, where $\mathbb{F}$ is a field. Show that $F(R)$(the field of fractions) may be identified with the ring $\mathbb{F}$$((x))$ of formal Laurent series. A formal Laurent ...
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0answers
95 views

Can we construct a $\mathbb Q$-basis for the Pythagorean closure of $\mathbb Q?$

This is a follow-up question to this one. I asked it there first but moved it here following the advice from Cam McLeman. I tried to prove that $(\mathbb P:\mathbb Q)=\aleph_0$ and I think I ...
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3answers
325 views

Is the Pythagorean closure of $\mathbb Q$ equal to the field of constructible numbers?

A Pythagorean field is one in which every sum of two squares is again a square. $\mathbb Q$ is not Pythagorean, which is easy to see. I have read a theorem online which says that every field has a ...
2
votes
2answers
225 views

Frobenius Auto need not be an automorphism if F is infinite

I'm trying to find an example to show the map $\sigma_p : F \rightarrow F$ given by $\sigma_p(a)=a^p$ for $a\in F$ need not be an automorphism in the case that F is infinite. I'm lost as to where to ...