Why is : $ [ \mathbb{R} [x_1 , \dots , x_n ]^{ A_n } : \mathbb{R} [x_1 , \dots , x_n ]^{ S_n } ] = [ \mathfrak{S}_n : \mathfrak{A}_n ] $?

Question

How to show that : $$ [ \mathbb{R} [x_1 , \dots , x_n ]^{ \mathfrak{A}_n } : \mathbb{R} [x_1 , \dots , x_n ]^{ \mathfrak{S}_n } ]= [ \mathfrak{S}_n : \mathfrak{A}_n ] $$ such that :

$ \mathfrak{S}_n $ : the symmetric group

$ \mathfrak{A}_n $ : the alternating subgroup of $ \mathfrak{S}_n $.

$ \mathbb{R} [x_1 , \dots , x_n ]^{ \mathfrak{S}_n } $ : the space of invariants polynomials under the action of : $ \mathfrak{S}_n $

$ \mathbb{R} [x_1 , \dots , x_n ]^{ \mathfrak{A}_n } $ : the space of invariants polynomials under the action of : $ \mathfrak{A}_n $

Thanks in advance for your help.