# Questions tagged [normal-extension]

For questions about normal field extensions.

89 questions
Filter by
Sorted by
Tagged with
52 views

### Why this extension is Galois?

Taken from Algebra : Chapter 0 by Paolo Aluffi "Let $f(x) \in \mathbb{C}[x]$ be a nonconstant polynomial; we have to prove that $f(x)$ has roots in $\mathbb{C}$. Note that if $f(x)$ has no roots ...
• 1
91 views

### When normal extensions are normal

I am wondering if the following statement is correct for each of the following properties $\mathcal P:$ normal, separable, and Galois or not: If $K \subseteq L \subseteq M$ are fields and $M/K$ has ...
• 2,069
25 views

### $E/F$ is normal iff $E$ is a splitting field of some $f(x)\in F[x]$, is it always valid?

This result is proven and well known for finite field extensions, however, consider the question: Let $F$ be a field and let $E/F$ be a finite extension. Suppose that $\alpha_1, \dots , \alpha_k \in E$...
• 931
20 views

• 2,043
1 vote
208 views

### Why isn’t $\mathbb{Q}[x]/\langle x^3-2\rangle$ a normal extension over $\mathbb{Q}$?

The extension $\mathbb{Q}[x]/\langle x^3-2\rangle$ is given on my lecture notes as an example of a non normal extension over $\mathbb{Q}$. I understand why $\mathbb{Q}[\sqrt[3]{2}]$ is not: because it ...
• 1,786
43 views

• 31
293 views

### Root of the polynomial $x^3-x-1$ over $\mathbb{Q}$ does not generate normal extension

Show that the field generated by a root of $f(x)$ where $f(x)=x^3-x-1$ over $\mathbb{Q}$ is not normal over $\mathbb{Q}$. First in order to mention some extension I showed $f(x)=x^3-x-1$ is ...
202 views

### Non-normal extension of Q

The counterexamples of normal extensions over $\mathbb{Q}$ that I could find are basically all in the form of $E = \mathbb{Q}(\sqrt[n]{\alpha})$ (with $x^n - \alpha$ being its minimal polynomial; and ...
53 views

### Find a particular intermediate field $M$ such that $\mathbb{Q}\subset M\subset\mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})$.

Problem: Let $\alpha=\sqrt{\frac{3+i\sqrt{7}}{2}}$ and $K=\mathbb{Q}(\alpha)$. Find the fixed field $M=\{x\in \mathbb{Q}(\sqrt{\frac{3+i\sqrt{7}}{2}})|\sigma(x)=x\}$, where $\sigma$ is the ...
• 1,457
1 vote
112 views

### Splitting fields are normal using symmetric polynomials.

I was told it is possible to prove splitting fields are normal using the Fundamental Theorem of Symmetric Polynomials rather than the usual approach. Does anyone have hints or a reference for this? ...
1 vote
82 views

### What does an embedding over $K$ mean?

I have some understanding problem. In the topic of extensions they always speak about embeddings over some field $K$ and i have absolutly no idea what this means. One example is the following: Let $K$...
• 2,018
173 views

### What does it mean that a normal extension remain normal under lifting?

I consider the book "Algebra" by Serge Lang and on page 238 he has the theorem 3.4 saying that normal extensions remain normal under lifting. I don't see what he means by that, and therefore ...
• 2,018
1 vote
59 views

### Can someone explain me what they did in the proof that every extension of degree 2 is normal?

I consider the following proof that every extension of degree $2$ is normal and I got stuck somewhere: Let $K$ be an extension of $F$ with $[K:F]=2$. Then $K=F(\alpha)$ where $\alpha$ is a root of an ...
• 2,018
1 vote
71 views

### Is $F\hat{E_s}/E$ normal, where $F/E$ is a finite field extension?

Let $F/E\,$ be a finite field extension, let $E_s$ be the separable closure of $E$ in $F$ and let $\hat{E_s}$ be its normal closure. Then $\hat{E_s}/E$ is Galois and $F\hat{E_s}/\hat{E_s}$ is purely ...
• 458
104 views

### On normal extension, a square root of discriminant is in field F?

The probrem I have now is Let $F$ be a Field, and $f(X) = X^3+aX+b \in F[X]$ be a irreducible polynomial. Also, $\alpha, \beta, \gamma \in E$ ($E$ is the smallest splitting field of $f$) are roots of ...
• 405
50 views

• 3,124
1 vote
132 views

### Normal closure as the compositum of conjugates

An excerpt from Serge Lang's Algebra Chapter V $\S4$ p. 242. Let $E$ be a finite extension of $k$. The intersection of all normal extensions $K$ of $k$ (in an algebraic closure $E^\text{a}$) ...
• 1,257