I have 4 questions for the same problem. I worked on this for days but after while, I didn't get very far.(adsbygoogle = window.adsbygoogle || []).push({});

Q1. To show that [itex]f(x)=x^4-2x^2-1[/itex] is irreducible, which irreducibility test is the most efficient? Reduction mod p doesn't really work (when p=2) and neither does Eisenstein's Irreducibility Criterion.

I supposed that f(x) can be written as a linear factor and a cubic polynomial and came up with a contradiction. I then supposed that f(x) can be written as a product of two quadratic polynomials and came with a contradiction. These processes work but they're just too long.

Any efficient irreducibility suggestions/tests for this particular polynomial: [itex]f(x)=x^4-2x^2-1[/itex]?

Q2. Let [itex]a= (1+\sqrt{2})^{1/2} [/itex] and [itex]b= (1-\sqrt{2})^{1/2} [/itex]. Does [itex]\mathbb{Q}(b)= \mathbb{Q}(a,i)[/itex]?

Q3. Let [itex]E=\mathbb{Q}(a)[/itex]. To find the index [itex] [E:\mathbb{Q}] [/itex], is this 4 or 8? I thought it would equal the degree of the irreducible polynomial for [itex]a[/itex], which is 4, but [itex]b[/itex] is also a root of the above polynomial and [itex]b[/itex] is a complex number, not real. So since we need to adjoin [itex]i[/itex], I thought the index should be 8.

Q4. What is the Galois group of E over [itex]\mathbb{Q}[/itex]? I found a similar problem from Dummit and Foote and its Galois group is [itex]\mathscr{D}_4[/itex] but someone mentioned that it should be [itex]\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2[/itex]. So which one is it and how does one know?

Thanks for your time!

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# Galois Theory (degree 4 polynomial)

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