Let \( E \subset F \) and \( F \subset K \) be field extensions which are not algebraic. Let \( \alpha \in K \) be algebraic over \( F \) and \( \alpha \notin F \). Let \( L \) be the subfield of \( K \) generated over \( E \) by the coefficients of the monic polynomial of minimal degree over \( F \) which has \( \alpha \) as a zero. Then, which of the following is/are TRUE?
Show Hint
- \( F(\alpha) \supset L(\alpha) \) is a finite extension if and only if \( F \supset L \) is a finite extension
- The dimension of \( L(\alpha) \) over \( L \) is greater than the dimension of \( F(\alpha) \) over \( F \)
- The dimension of \( L(\alpha) \) over \( L \) is smaller than the dimension of \( F(\alpha) \) over \( F \)
- \( F(\alpha) \supset L(\alpha) \) is an algebraic extension if and only if \( F \supset L \) is an algebraic extension
The Correct Option is A, D
Solution and Explanation
Since \( \alpha \) is algebraic over \( F \), both \( F(\alpha) \) and \( L(\alpha) \) are algebraic extensions. Therefore, the finiteness of the extension is preserved between the fields, making (A) TRUE.
Step 2: The dimension of \( L(\alpha) \) over \( L \) is greater than the dimension of \( F(\alpha) \) over \( F \)
Since \( L \subset F \) and \( L(\alpha) \) is generated by a minimal polynomial over \( L \), the dimension of \( L(\alpha) \) over \( L \) cannot be greater than that of \( F(\alpha) \) over \( F \). Hence, (B) is FALSE.
Step 3: The dimension of \( L(\alpha) \) over \( L \) is smaller than the dimension of \( F(\alpha) \) over \( F \)
This is not necessarily true either, as \( \alpha \) might have the same minimal polynomial over both \( F \) and \( L \). Therefore, (C) is FALSE.
Step 4: \( F(\alpha) \supset L(\alpha) \) is an algebraic extension if and only if \( F \supset L \) is an algebraic extension
Since \( \alpha \) is algebraic over \( F \), both \( F(\alpha) \) and \( L(\alpha) \) are algebraic extensions. However, this doesn't imply that \( F \supset L \) is algebraic unless further conditions are given. So (D) is also NOT always TRUE. Hence, (D) is FALSE.
Final Answer
\[ \boxed{(A) \ F(\alpha) \supset L(\alpha) \text{ is a finite extension if and only if } F \supset L \text{ is a finite extension}} \]
Top Questions on Subsets
- All rings considered below are assumed to be associative and commutative with \( 1 \neq 0 \). Further, all ring homomorphisms map 1 to 1.
Consider the following statements about such a ring \( R \):
P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).
Then, which of the following is/are TRUE? - Let \( X \) be an uncountable set. Let the topology on \( X \) be defined by declaring a subset \( U \subset X \) to be open if \( X - U \) is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?
- In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation \( \sim \) on the sphere \[ S = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \right\} \] by \( (x_1, x_2, x_3) \sim (y_1, y_2, y_3) \) if \( x_3 = y_3 \), for \( (x_1, x_2, x_3), (y_1, y_2, y_3) \in S \). Let \( [x_1, x_2, x_3] \) denote the equivalence class of \( (x_1, x_2, x_3) \), and let \( X \) denote the set of all such equivalence classes. Let \( L : S \to X \) be given by \[ L\left( (x_1, x_2, x_3) \right) = [x_1, x_2, x_3]. \] If \( X \) is provided with the quotient topology induced by the map \( L \), then which one of the following is TRUE?
Let the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements:
S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact.
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
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“I put the brown paper in my pocket along with the chalks, and possibly other things. I suppose every one must have reflected how primeval and how poetical are the things that one carries in one’s pocket: the pocket-knife, for instance the type of all human tools, the infant of the sword. Once I planned to write a book of poems entirely about the things in my pocket. But I found it would be too long: and the age of the great epics is past.” (From G.K. Chesterton’s “A Piece of Chalk”)
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