Can someone help me understand how the splitting field of x2 - 2 over Q (the rationals) is Q(√2, √(2)e2πi/3)? In general, I am confused as to why/how we get e(p/q)πi in certain extension fields. Admittedly, I don't really know what I'm talking about, so my question may not really make sense, but it would be appreciated if someone could give an explanation of splitting fields. Thanks a lot.
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Explaination of Splitting Fields?
- Thread starter JonJon
- Start date
Can someone help me understand how the splitting field of x2 - 2 over Q (the rationals) is Q(√2, √(2)e2πi/3)? In general, I am confused as to why/how we get e(p/q)πi in certain extension fields. Admittedly, I don't really know what I'm talking about, so my question may not really make sense, but it would be appreciated if someone could give an explanation of splitting fields. Thanks a lot.
There is not just one explicit notation for a splitting field, but it is common to write the splitting field of \(\displaystyle x^n-a\in \mathbb{Q}[x]\) as \(\displaystyle \mathbb{Q}(\xi, \sqrt[n]{a})\) where \(\displaystyle \xi\) is a primitive nth root of unity.
That is not the splitting field for your polynomial. Because in this case, the second roots of unity are just 1 and -1. So your splitting field is just \(\displaystyle \mathbb{Q}(\sqrt{2})\)
O.K, so I guess my question is where/why/how does the primitive nth root of unity come into play?
The roots of the polynomial \(\displaystyle x^n-a\in \mathbb{Q}[x]\) are \(\displaystyle S=\{\xi^k\sqrt[n]{a};\,\, 0\le k < n\}\) . Normally \(\displaystyle \xi\) is taken to be the first complex nth root of unity: \(\displaystyle \xi = e^{2\pi i/n}\) (assuming n>2, it is not real).
The splitting field for any polynomial, is the base field adjoined with the roots of the polynomial. So the splitting field for \(\displaystyle x^n-a\) is: \(\displaystyle \mathbb{Q}(S)\)
But it is just simplified noting that equality can be shown, i.e., \(\displaystyle \mathbb{Q}(S) = \mathbb{Q}(\sqrt[n]{a}, \xi)\)