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Dummit and Foote Exercise 1 on page 545 reads as follows:
Determine the splitting field and its degree over \mathbb{Q} for x^4 - 2.
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I have started on the solution to this exercise as follows:
The two roots of x^4 - 2 are \alpha = \sqrt[4]{2} and \beta = \sqrt[4]{2}i
Thus the splitting field is \mathbb{Q}(\sqrt[4]{2}, \sqrt[4]{2}i )
We note that $$ \mathbb{Q}(\sqrt[4]{2}, \sqrt[4]{2}i ) = \mathbb{Q}(\sqrt[4]{2}, i ) $$ since the product of $$ \sqrt[4]{2} $$ and i must be in $$ \mathbb{Q}(\sqrt[4]{2}, i ) $$The element \alpha = \sqrt[4]{2} \in \mathbb{R} is algebraic over \mathbb{Q} and so m_{\alpha}(x) = m_{\sqrt[4]{2}}(x) = x^4 - 2 is the minimal polynomial for \alpha = \sqrt[4]{2} over \mathbb{Q}.
Thus the degree of \alpha = \sqrt[4]{2} over \mathbb{Q} is the degree of m_{\alpha}(x) = 4
Hence $$ [\mathbb{Q}(\sqrt[4]{2}) \ : \ \mathbb{Q}] $$ = 4
The element \beta = \sqrt[4]{2}i \in \mathbb{C} is also algebraic over \mathbb{Q} and so m_{\beta}(x) = m_{\sqrt[4]{2}i}(x) = x^4 - 2 is the minimal polynomial for \beta = \sqrt[4]{2}i over \mathbb{Q}.
Thus the degree of \beta = \sqrt[4]{2}i over \mathbb{Q} is the degree of m_{\beta}(x) = 4
Hence $$ [\mathbb{Q}(\sqrt[4]{2}i) \ : \ \mathbb{Q}] $$ = 4
But where to from here - need to find the degree of the splitting field.
Can someone please confirm that my reasoning above is valid and show me the way forward from here?
Peter[Note that this has also been posted on MHF]
Determine the splitting field and its degree over \mathbb{Q} for x^4 - 2.
------------------------------------------------------------------------------------------------------
I have started on the solution to this exercise as follows:
The two roots of x^4 - 2 are \alpha = \sqrt[4]{2} and \beta = \sqrt[4]{2}i
Thus the splitting field is \mathbb{Q}(\sqrt[4]{2}, \sqrt[4]{2}i )
We note that $$ \mathbb{Q}(\sqrt[4]{2}, \sqrt[4]{2}i ) = \mathbb{Q}(\sqrt[4]{2}, i ) $$ since the product of $$ \sqrt[4]{2} $$ and i must be in $$ \mathbb{Q}(\sqrt[4]{2}, i ) $$The element \alpha = \sqrt[4]{2} \in \mathbb{R} is algebraic over \mathbb{Q} and so m_{\alpha}(x) = m_{\sqrt[4]{2}}(x) = x^4 - 2 is the minimal polynomial for \alpha = \sqrt[4]{2} over \mathbb{Q}.
Thus the degree of \alpha = \sqrt[4]{2} over \mathbb{Q} is the degree of m_{\alpha}(x) = 4
Hence $$ [\mathbb{Q}(\sqrt[4]{2}) \ : \ \mathbb{Q}] $$ = 4
The element \beta = \sqrt[4]{2}i \in \mathbb{C} is also algebraic over \mathbb{Q} and so m_{\beta}(x) = m_{\sqrt[4]{2}i}(x) = x^4 - 2 is the minimal polynomial for \beta = \sqrt[4]{2}i over \mathbb{Q}.
Thus the degree of \beta = \sqrt[4]{2}i over \mathbb{Q} is the degree of m_{\beta}(x) = 4
Hence $$ [\mathbb{Q}(\sqrt[4]{2}i) \ : \ \mathbb{Q}] $$ = 4
But where to from here - need to find the degree of the splitting field.
Can someone please confirm that my reasoning above is valid and show me the way forward from here?
Peter[Note that this has also been posted on MHF]
Last edited: