Linked Questions

6
votes
1answer
1k views

Galois group of a biquadratic quartic

From Hungerford, section V, chapter 4 exercise 9: Let $x^4+ax^2+b$ in $K[x]$ (with char $K\neq $2) be irreducible with Galois group $G$. (a) If $b$ is a square in $K$, then $G = \mathbb{Z}...
1
vote
1answer
656 views

The Galois Group of $x^4 - 5x^2 + 6$ [duplicate]

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
0
votes
1answer
105 views

Determine the Galois group of $X^4 - 2X^2 +2$ over $\Bbb Q$ and prove $\sqrt 2 \in \Bbb Q(\alpha)$ where $\alpha$ is a root of $X^4 - 2X^2 +2$ [duplicate]

Determine the Galois group of $X^4 - 2X^2 +2$ over $\Bbb Q$ EDIT : I want to address the particular polynomial and not the full general case. In order to solve this problem, I want to have a self-...
19
votes
1answer
7k views

Galois group of $X^4 + 4X^2 + 2$ over $\mathbb Q$.

I'd like to calculate the Galois group of the polynomial $f = X^4 + 4X^2 + 2$ over $\mathbb Q$. My thoughts so far: By Eisenstein, $f$ is irreducible over $\mathbb Q$. So $\mathrm{Gal}(f)$ must be a ...
9
votes
4answers
1k views

Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$

I would like to know how to solve part $ii)$ of the following problem: Let $K /\mathbb{Q}$ be a splitting field for $f(X) =X^4-3X^2+5$. i) Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ ...
3
votes
2answers
1k views

computing the galois group of a polynomial

Compute the Galois group of the splitting field of the polynomial $t^4-3t^2+4$ over $\mathbb{Q}$. I don't know how can I do this problem, the roots are very "ugly" maybe if I consider another basis (...
3
votes
3answers
1k views

Galois group of $f(X) = X^4 + 3X^2 + 1$

I'm trying to compute the Galois group $G$ of $$f=X^4 + 3X^2 + 1$$ over $\mathbb{Q}$. This is what I worked out so far: The discriminant is $2^4\cdot 5^2$, a square, so $G\subset A_4$ $f$ is ...
5
votes
1answer
730 views

Galois group of $X^4 + 2X^2+4$

Find the Galois group of $f(X) = X^4 + 2X^2+4$ over $\mathbb{Q}$. Let $L$ be the splitting field of $f$ over $\mathbb{Q}$. Finding the roots of this polynomial, I got $$X^2 = \frac{-2\pm \sqrt{4-16}}...
4
votes
1answer
622 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm \sqrt{\...
0
votes
1answer
340 views

Find the Galois group of $x^4-x^2-6$.

I'm trying to find the Galois group of $x^4-x^2-6$. I think there are 4 roots, thus I guess the Galois group is $A_4$? But I don't know in general how to solve this.
3
votes
0answers
412 views

Galois group of $x^4-2x^2+2$. Apparent contradiction!!

I'm trying to compute de Galois group of $p(x) = x^4-2x^2+2$, and I seem to have arrived to a contradiction, which of course means that there is some mistake that I'm not able to find. The first ...
2
votes
0answers
323 views

Finding the Galois group of $x^4+5x^2+5$

Find the Galois group of $f(x)=x^4+5x^2+5\in \mathbb{Q}[x]$. This is solved here, Exersice 3: https://math.berkeley.edu/~serganov/114/solhwg.pdf I have a question about it (I will not write all the ...
4
votes
0answers
269 views

Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$ Step 1: Calculate cubic resolvent Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent Step 3: If $\...
1
vote
0answers
68 views

Degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ and splitting field

I have two questions: Determine the degree of the extension degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ over $\Bbb Q$ and the degree of the splitting field of the minimal polynomial of $\sqrt{2 + \...