Mark Stewart
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Just working on some Galois Theory questions (from Stewart’s book) and wanting to know what others get…?
Mark the following true or false.
a) Extensions of the same degree are isomorphic
b) Isomorphic extensions have the same degree
c) Every algebraic extension is finite
d) Every transcendental extension is not finite
e) Every element of C is algebraic over R
f) Every extension of R is finite
g) Every algebraic extension of Q is finite
h) Every vector space is isomorphic to the vector space corresponding to some field extension
i) Every extension of a finite field is finite
a false counterexample Q[sqrt(2)] and Q[sqrt(3)]
b true
c false algebraic closure of Q is algebraic but not finite
d true
e true, it satisfies some quadratic equation
f if you mean every algebraic extension is finite, that’s true otherwise it’s false
g false, the same example as in c
h true Just add the suitable root of unity
i false algebraic closure of Zp(a field with p elements, p is prime number) is not finite
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