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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9
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874 views

Showing a homomorphism of a field algebraic over $\mathbb{Q}$ to itself is an isomorphism.

Suppose $F$ is algebraic over $\mathbb{Q}$ and $\varphi : F\to F$ is a homomorphism. Prove $\varphi$ is an isomorphism. Showing injectivity follows from the fact that the only ideals in a field are $...
5
votes
1answer
775 views

Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$

Following my previous question the book then asks Use this to determine when the field extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$ is biquadratic (where $a,b\in\mathbb{Q}$) "...
4
votes
2answers
350 views

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$ for some $n$, where $\zeta_n=e^{2\pi i/n}$? I think the answer is no, but I can't give a full proof. Assume the contrary, we then have $\mathbb{Q}(...
3
votes
3answers
2k views

Extensions of degree two are Galois Extensions.

This question from Allan Clark's "Elements of Abstract Algebra" Show that an extension of degree 2 is Galois except possibly when the characteristic is 2. What is the case when the characteristic is ...
3
votes
3answers
489 views

Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
6
votes
2answers
660 views

Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.
3
votes
1answer
95 views

Fraction field of $F[X,Y](f)$ isomorphic to $F(X)[Y]/(f)$

Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$. Then, by Gauss's lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field (...
3
votes
1answer
161 views

Degree of splitting field less than n!

I've been asked to prove that if a polynomial $f\in \mathbb{Q}[x]$ has degree $n$, then the splitting field of $f$ has degree less than or equal to $n!$. That is, $[\mathbb{Q}(\alpha_1,...,\alpha_n):\...
3
votes
1answer
84 views

Commutativity of “extension” and “taking the radical” of ideals

Let $K$ be a field (not necessarily algebraically closed) and $\overline{K}$ its algebraic closure. By $K[\text{X}]$, I mean $K[X_1,...,X_n]$. Is it true that the operations of "extension" and "...
3
votes
3answers
282 views

Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?

As title, Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated? Say if $K/F$ is a finite extension when $K$ is a finite-dimensional vector space of $F$. Clearly, ...
2
votes
1answer
95 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
0
votes
2answers
108 views

Number of solutions for the given equation in finite field of order 32.

Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$. I have figured out that non zero elements of this field forms a cyclic multiplicative group of ...
3
votes
2answers
224 views

Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?

I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$. I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
3
votes
1answer
112 views

Embedding Fields in Matrix Rings

Is well known that the field $\mathbb C$ of complex numbers can be embedded in the ring $M_2(\mathbb R)$ of matrices of order two over de reals. In fact, $\varphi :\mathbb C\longrightarrow M_2(\mathbb ...
26
votes
1answer
425 views

Rigidity of the category of fields

Let's call a category rigid if every self-equivalence is isomorphic to the identity. For example, $\mathsf{Set}$, $\mathsf{Grp}$, $\mathsf{Ab}$, $\mathsf{CRing}$ (MO/106838), $\mathsf{Top}$ (SE/450193)...
32
votes
4answers
2k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
22
votes
2answers
1k views

Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?

This question is taken from Dummit and Foote (14.9 #6). Any help will be appreciated: Show that if $t$ is transcendental over $\mathbb{Q}$, then $\mathbb{Q}(t,\sqrt{t^3-t})$ is not a purely ...
15
votes
2answers
2k views

Basis of primitive nth Roots in a Cyclotomic Extension?

While reading one of Keith Conrad's great blurbs, Linear Independence of Characters, there is a footnote at the bottom of page 2 saying In general, the primitive $n$th roots of unity in the $n$th ...
12
votes
2answers
2k views

Why does an irreducible polynomial split into irreducible factors of equal degree over a Galois extension?

I've been struggling to prove this fact over the past day or so. Suppose $f(x)\in F[X]$ is irreducible over a field $F$ with $\deg(f)=n$, and let $L$ be the splitting field of $f(x)$ over $F$ with ...
15
votes
1answer
396 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an example?...
15
votes
5answers
462 views

Why does $K \leadsto K(X)$ preserve the degree of field extensions?

The following is a problem in an algebra textbook, probably a well-known fact, but I just don't know how to Google it. Let $K/k$ be a finite field extension. Then $K(X)/k(X)$ is also finite with ...
12
votes
1answer
259 views

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite?

Is there a proper subfield $K\subset \mathbb R$ such that $[\mathbb R:K]$ is finite? Here $[\mathbb R:K]$ means the dimension of $\mathbb R$ as a $K$-vector space. What I have tried: If we can find ...
28
votes
3answers
2k views

Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. ...
24
votes
6answers
1k views

Proof of Existence of Algebraic Closure: Too simple to be true?

Having read the classical proof of the existence of an Algebraic Closure (originally due to Artin), I wondered what is wrong with the following simplification (it must be wrong, otherwise why would we ...
6
votes
2answers
619 views

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$?

Is $\mathbf{Q}(\sqrt{2},\sqrt{3}) = \mathbf{Q}(\sqrt{6})$? If $\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in\mathbf{Q}\}$ and $\mathbf{Q}(\sqrt{6})= \{a+b\sqrt{6} | ...
6
votes
2answers
234 views

Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?

I heard that $\Bbb R$ and $\Bbb C$ are the only connected, locally compact fields. Does anyone know a proof for this result?
10
votes
2answers
498 views

Necessarily a field between a field and its algebraic extension

This is an exercise in some textbooks. Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field. The trouble is really with ...
9
votes
1answer
472 views

For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$ [closed]

Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that strictly contains $F$. Why is $x$ algebraic over $K$? Thanks a lot!
8
votes
3answers
2k views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
8
votes
5answers
9k 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 $x^2 + 1$....
5
votes
2answers
153 views

Finding a fixed subfield of $\mathbb{Q}(t)$

I'm looking at automorphisms $t \to 1-t$ and $t \to \frac{1}{t}$ of the field $\mathbb{Q}(t)$. By looking at the relations between these I think I've found the group generated by them to be $S_3$. ...
4
votes
4answers
3k views

Examples of fields of characteristic 0?

I was preparing for an area exam in analysis and came across a problem in the book Real Analysis by Haaser & Sullivan. From p.34 Q 2.4.3, If the field F is isomorphic to the subset S' of F', show ...
6
votes
2answers
648 views

degree of a field extension

Let $\alpha$ be a root of $x^3+3x-1$ and $\beta$ be a root of $x^3-x+2$. What is the degree of $\mathbb{Q}(\alpha^2+\beta)$ over $\mathbb{Q}$? My guess is 9, because i found a monic polynomial of ...
6
votes
4answers
224 views

Number fields with all degrees equal to a power of three

Say that a number field $\mathbb K$ is $3$-powerful if the degree (over $\mathbb Q$) of every non-rational element of ${\mathbb K}$ is a power of $3$. By Zorn’s lemma, the field $\cal A$ of all ...
4
votes
2answers
1k views

isomorphism of Dedekind complete ordered fields

In the last chapter of Spivak's Calculus, there is a proof that complete ordered fields are unique up to isomorphism. I find the first steps in it somewhat suspicious. Specifically, I believe he is ...
3
votes
2answers
619 views

A question about a weak form of Hilbert's Nullstellensatz

Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows: Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$. We know ...
3
votes
1answer
235 views

Fixed Field of Automorphisms of $k(x)$

Fixed field of automorphisms of $k(x)$, with $k$ a field, induced by $I(x)=x$, $\varphi_1(x) = \frac{1}{1-x}$, $\varphi_2 (x)=\frac{x-1}{x}$? Since $I(x)=x$, $\varphi_1(x)=\frac{1}{1-x}$, $\varphi_2 ...
5
votes
2answers
1k views

Galois Field Fourier Transform

there are two definitions for Reed-Solomon codes, as transmitting points and as BCH code ( http://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction ). On Wikipedia there is written that we ...
6
votes
1answer
936 views

field of algebraic numbers

Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals? Many Thanks
6
votes
3answers
639 views

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 \...
3
votes
2answers
98 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 ...
1
vote
1answer
129 views

Show that $9+9x+3x^3+6x^4+3x^5+x^6$ is irreducible given one of its roots

Given a polynomial $f(x)=9+9x+3x^3+6x^4+3x^5+x^6$ and one of its roots $\alpha=2^{1/3}+e^{2\pi i/3}$. Show that $f(x)$ is irreducible in $\mathbb Q[x]$. Eisenstein's criterion fails, it also didn't ...
1
vote
2answers
758 views

An example for a homomorphism that is not an automorphism

Let $K/F$ be a field extension, I know that if $K/F$ is a finite extension then a simple argument from linear algebra shows that since every homomorphism of fields from $K$ to $K$ that fixes $F$ is 1-...
1
vote
1answer
77 views

Convert from a field extension to an elementary field extension

I have a basic question about algebraic field extensions: How can I convert a multiple extension like $\mathbb{Q}(\sqrt{2},\sqrt{3})$ to a single (elementary) field extension (like $\mathbb{Q}(\sqrt{...
7
votes
1answer
2k views

Sums and products of algebraic numbers

How does one go about proving that the sums and products of two algebraic numbers over a field $F$ (say $a,b\in K$, where $K/F$ is a field extension) is also algebraic? If we call $f_a$ and $f_b$ ...
5
votes
2answers
175 views

Closure of a number field with respect to roots of a cubic

Consider the following property of subfields ${\mathbb K}$ of ${\mathbb C}$ : $$ \text{Any polynomial of degree 3 with coefficients in} \ {\mathbb K} \ \text{has a root in } {\mathbb K} \ \ \ \ (*)...
5
votes
4answers
465 views

Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing

I need to rationalize $\displaystyle\frac{1}{a+b\sqrt[3]2 + c(\sqrt[3]2)^2}$ I'm given what I need to rationalize it, namely $\displaystyle\frac{(a^2-2bc)+(-ab+2c^2)\sqrt[3]2+(b^2-ac)\sqrt[3]2^2}{(a^...
5
votes
1answer
335 views

What is the meaning of $\mathbf{Q}(\sqrt{2},\sqrt{3})$

I know that : $\mathbf{Q}(\sqrt{2}) = \mathbf{Q}+ \sqrt{2} \mathbf{Q}$ , but then what is $\mathbf{Q}(\sqrt{2},\sqrt{3})$?
4
votes
1answer
911 views

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$

Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$. My justification for this question is as follows; Suppose $F(\alpha^2)\subsetneq F(\alpha)$, we have $F \subsetneq F(\alpha^2) \...
3
votes
1answer
127 views

Polynomials having as roots the sum (respectively, the product) of two algebraic elements

This question remained somehow incompletely solved. The OP also asked for an explicit form of the minimal polynomial of the sum (respectively, product) of two algebraic elements (in a field extension)....